210HWChapter05

210HWChapter05 - a) Experiment = play the game or test the...

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Unformatted text preview: a) Experiment = play the game or test the game b) 75 of the veteran game players said that it had bad playability 65 is not a probability because the number 65 is not between 0 c) and 1, inclusive. d) The number -1 is not between the numbers 0 and 1, inclusive. e) After the tests were all done, 54 said they did not like it count said liked it empirical or relative frequency 80 65 0.8125 1 a) Total cards # of queens in 1 deck Probability of selecting a Queen 52 4 0.0769230769 Classical because there are n equally likely outcomes b) 2 a) Total Children Marital Status of Parents Divorced Married 539 333 P(Married) = P(Divorced) = 0.6178107607 Widowed 182 24 P(Widowed) = 0.3376623377 0.0445269017 Empirical Probability or Relative Frequency because the problem uses past date to calculate a probability that could be used for future estimation. Law of Large Numbers says that as the number of b) Trials the Relative Frequency will approach the true probability. 3 a) 1 because it is above 12,000. The book was written a while ago. I guess it could be either classical or empirical because 1/1 = 1 ==> 100%. You could say its likelihood is 1, but in some ways it could be argued that it is none of the b) approaches to probability because it is one, known event from the past. Equal? TRUE Two People = Favor = F Against = A Possible Outcomes Outcome 1 F, F Outcome 2 F, A Outcome 3 A, A Outcome 4 A, F 2 Majors Acc Fin Econ Manage Market Total a) 34 Prob Name Probability Number P(Acc) = 0.2941176471 P(Fin) = 0.1470588235 P(Econ) = 0.0882352941 P(Manage) 0.1764705882 = P(Market) = 0.2941176471 Total b) # of Students Declaring Major 10 5 3 6 10 1 This is the Relative Frequency or Empirical Approach Total Candidates Minority a) b) P(Minority) = Classical because there are n equally likely outcomes 5 2 0.4 a) b) c) d) Relative Frequency or Empirical Classical Classical, but still extremely remote chance of winning Empirical based on past data. Sample size = People asked Yes or No question: 40 Oil Executives The experiment was the asking the 40 executives a) the Yes/No environmental question b) All 40 said No. c) Answered Yes P(Yes) = 10 d) Relative Frequency or Empirical e) No, not both. Each of the possible outcomes are not equally likely. For example ==> 0.25 1 possible outcome is this many said yes: 2 possible outcome is this many said yes: 12 P(Yes) = 0.3 The possible outcomes are mutually exclusive because you can't have the group have 12 say yes and 19 say yes also. 19 P(Yes) = 0.475 Number of Violations Number of Drivers 0 1,910 1 46 2 18 3 12 4 9 5 or more 5 Total 2,000 The experiment is seeing how many speeding violations people have from a sample of 2000 people. One possible event is 1 speeding violation. Another example of b) one possible event is one or more violations. c) 0.009 d) Empirical or Relative Frequency a) Table of possible probabilities P(exactly 0 speeding violations) = P(exactly 1 speeding violations) = P(exactly 2 speeding violations) = P(exactly 3 speeding violations) = P(exactly 4 speeding violations) = P(5 or more speeding violations) = P(3 or more) = 0.006 + 0.0045 + 0.0025 = 0.955 0.023 0.009 0.006 0.0045 0.0025 0.013 c7466754d66532ff4fbbea246d745bff7605e657.xls - SR 5-3 Classifications Event Supervisors A Maintenance B Production C Management D Secretarial E # of Employees 120 50 1460 302 68 2000 P(X) P(A) P(B) P(C) P(D) P(E) P(X) 0.06 0.025 0.73 0.151 0.034 1 a) i P(Maintenance or Secretarial) ii P(Not Management) = P(B or E) = = P(Not D) = b) Management Maintenance Secretarial c) (a)(i) P(B) P(E) Total Mutually Exclusive and not complimentary (not complimentary because they don't add up to 1) 0.025 0.034 0.059 0.025 + 1- 0.034 = 0.151 = 0.059 0.849 Need corrective shoes Need major dental work Percent that need corrective shoes and major dental work BOTH a) P(Need corrective shoes OR Need major dental work) = 0.15 + 0.08 - 0.03 = 8.00% 15.00% 3.00% Must subtract so you don't double count the 3.00% is BOTH 20.00% b) Need corrective shoes Need major dental work c7466754d66532ff4fbbea246d745bff7605e657.xls - Ex13 Income After Taxes Number of Firms Event Under $1 million 102 A $1 million to $20 million 61 B $20 million or more 37 C 200 P(X) P(A) P(B) P(C) P(X) 0.51 0.305 0.185 1 a) P(A) 0.51 P($1 million to $20 million or $20 million or b) more) = 0.3050 + 0.1850 0.0000 = 0.4900 Special Rule because the categories are mutually exclusive. However, if you used the General Rule, the joint probability would have been zero and you would have gotten the correct answer. Profit Break Even Lose Total a) b) P(Profit OR Break Even) = 0.5 + 0.3 = P(Profit OR Break Even) = 1 - 0.2 = 50% 30% 20% 100% 80% 80% A B 0.25 0.5 P( above C) =0.25 + 0.5 = 0.75 Two Coins are tossed A= B= H,H T,T P(H,H) = P(T,T) = Total 0.25 0.25 0.5 The two are mutually exclusive because the two events cannot happen at the same time. Two events, A & B, are mutually exclusive if both events, A & B, cannot occur at the same time a) b) No, because they do not add up to 1, they add up to .5 All Possible Outcomes 1 H,H 2 H,T 3 T,T 4 T,H Total 4 0.25 0.25 0.25 0.25 1 Events A B A AND B P(A OR B) = 0.2 + 0.3 - 0.15 = P(X) P(X) P(A) = 0.2 P(B) = 0.3 P(A AND B) = 0.15 0.35 P F D P&F&D P&D F&D P&F a) b) c) d) Southwest Grocery Stores Had Pharmacy Had Floral Shop Had Deli Had All 3 Had Pharmacy and Deli Had Floral Shop and Deli Had Pharmacy and Floral P(Had Pharmacy and Had Floral Shop) = P(Had Pharmacy and Had Deli) = No. Because, for example, if you select a store with a deli, it could be in a category "Deli only" or "Deli and Pharmacy". Joint Probability 0.4 0.5 0.7 0.1 0.3 0.25 0.2 P(F) = 0.5 0.2 0.3 P(D) = 0.7 e) Step 1: Find P(P or F or D) Step 2: Find 1 - P(P or F or D) = Notes: This is to add the Probability This is to not double count This if for the triple count Once you have all that you can subtract it from 1 P(P) = 0.4 P(P OR F OR D) = P(P) + P(F) + P(D) - P(P & D) P(F & D) - P(P & F) + P(P & F & D) 0.05 0.95 =C2+C3+C4-C6-C7-C8+C5 P(P) + P(F) + P(D) - P(P & D) - P(F & D) - P(P & F) + P(P & F & D) 1 - 0.95 = 0.05 P(P) = 0.4 P(F) = 0.5 P(D) = 0.7 This problem was not assigned Vacationers Going To Rocky Mountain Region Yellowstone 0.5 Tetons 0.4 Yellowstone and Tetons 0.35 P(Y) = P(T) = P(Y and T) = 0.5 + 0.4 - 0.35 = P(Y and T) = 0.5 + 0.4 - 0.35 = 0.35 0.50 0.40 0.55 a) P(Y and T) = 0.5 + 0.4 - 0.35 = 0.55 Notes: P("visit at least one" means Y or T or Both) But you can't double count b) "Joint" or "Concurrent" No because of the overlapping circles c) in the Venn Diagram to the right ==> P(Y) = Vacationers destinations according to National Park Service P(X) Compliment Yellowstone 0.5 0.5 Tetons 0.4 0.6 Yellowstone and Tetons 0.35 Visit only Yellowstone (0.5*(1-0.4)) Visit only Tetons (0.4*(1-0.5)) Yellowstone and Tetons 0.3 0.2 0.35 Done with formulas that take more time than they are worth P(Y) = P(T) = P(Y and T) = 0.50 0.40 0.55 P(T) = P(Y and T) = 0.5 + 0.4 - 0.35 = 0.35 a) 4/52 = 2/26 = 1/13 ==> because the sample space has not changed b) 3/51 because there are only 3 kings and the sample space has changed c) P(2 straight Kings) = 4/52*3/51 = 0.0045249 4 52 3 51 a) b) c) d) Derreck Lee Highest Ave in 2005 Season 0.335 Assume he hits in one game = 3 Empirical because it is based on past data and we are using it to estimate how he will do for the game P(3 straight hits ) = 0.335*0.335*0.335 = 0.037595375 0.0375954 P(No Hits) = (1-0.335)*(1-0.335)*(1-0.335) = (1-0.335)^3 = 0.294079625 0.2940796 P(at least 1 hit) = P(1 or 2 or 3 hits) = 1 - P(No Hits) = 0.7059204 or d) P(1) = P(1 hit, No hit, No Hit) + P(No hit, 1 Hit, No Hit) + P(No hit, No Hit, 1 hit) = P(2) = P(1 hit, 1 hit, No Hit) + P(1 Hit, No Hit, 1 hit) + P(No hit, 1 hit, 1 hit) = P(3) = P(1 hit, 1 hit, 1hit) = P(at least 1 hit) = P(1 or 2 or 3 hits) = 0.4444361 0.2238889 0.0375954 0.7059204 H NH DL Average = Probability of getting a hit DL Average = Probability of NOT getting a hit # of at bats = # of Trials = Because averages do not include walks, hit by pitch, sacrifices, and other non-Hits/NotHits in the calculation, The events H and NH are mutually exclusive and independent and the categories H and NH are collectively exhaustive. a b c d Empirical P(H and H and H) = P(H)*P(H)*P(H) = P(0) P(at least one hit) = P(1 or 2 or 3) 0.335 P(H) 0.665 P(NH) 3 0.037595375 0.294079625 0.705920375 check check 0.0376 0.0376 0.29408 0.70592 # of Possible outcomes in the sample space = ss = Possible Outcomes # of at bats = # of Trials = 1 2 1H H H 2H H NH 3H NH H 4 NH H H 5H NH NH 6 NH H NH 7 NH NH H 8 NH NH NH 8 3 # of Hits Probability out of 3 at of bats Occurance 3 2 2 2 1 1 1 0 0.03759538 0.07462963 0.07462963 0.07462963 0.14814538 0.14814538 0.14814538 0.29407963 1 Random Variable = X = # of Hits in 3 at bats P(X) 0 P(0) 1 P(1) 2 P(2) 3 P(3) P(X) 0.2940796 0.4444361 0.2238889 0.0375954 1 Teton Tires Failure hurdle P(XB-70 tire will last 60,000 before it becomes bald or fail) An adjustment is made on any tire that does not last 60,000 We purchase this # of tires: P(All 4 tires will last 60,000 miles) = 0.8*0.8*0.8*0.8 = 0.8^4 = 60,000 0.8 4 0.4096 Board of Directors Men Women Total Wee need to select a committee with 4 members at random 8 4 12 4 Sample Space Conditional Women Selected Changes Probability 4 12 0.3333333333 3 11 0.2727272727 2 10 0.2 1 9 0.1111111111 a) b) c) P(All Women) = 8/12*7/11*6/10*5/9 = 0.141414141414141 = P(All Men) = 4/12*3/11*2/10*1/9 = 0.00202020202020202 = No because 0.141414141414141 + 0.00202020202020202 Does not equal 1. In addition, there are other possibilities such as, for example, 3 women and a man. Also, the categories are not collectively exhaustive. Men Selected Sample Space Conditional Changes Probability 8 12 0.6666666667 7 11 0.6363636364 6 10 0.6 5 9 0.5555555556 0.1414141414 0.0020202020 Executives were asked whether or not they would remain with the company if they received a better offer from a different company Length of Service Less than 1-5 More than Loyalty 1 year years 6-10 years 10 years Total Would remain 10 30 5 75 120 Would not remain 25 15 10 30 80 35 45 15 105 200 P(Randomly selecting executive who has More than 10 years) = a) 105/200 = P(Randomly selecting executive who Would not remain GIVEN b) THAT More than 10 years) = 30/105 = P(Randomly selecting executive who has More than 10 years OR c) Would not remain) = 105/200 + 80/200 - 30/200 = P(Randomly selecting executive who Would remain AND who has More than 10 years) = P(Randomly selecting executive who Would not remain AND who has More than 10 years) = 0.525000 0.285714 0.775 0.375 0.375 0.15 0.15 Visits Yes Often Occasional Never Total a) b) Visits Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Often Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Occasional Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Never Enclosed Mall No Total 60 20 80 25 35 60 5 50 55 90 105 195 Because Independence means that the occurrence of one event does not affect the occurrence of another event, for our two variables "Visits" and "Enclosed Mall", they are not independent. Using the Rule of Independence, we can show that P(Never|Yes) Does Not Equal P(Never) ==> 5/90 Does Not Equal 55/195 Enclosed Mall Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No No No No No No No No No No No No No No No No No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No Yes Yes Yes Yes Yes No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No Enclosed Mall Visits Often Occasional Never Grand Total Data Yes CondProb Count of Visits 2/ 3 5/ 18 1/ 18 1/ 1 4/ 13 5/ 39 1/ 39 6/ 13 JointProb 30.77% 12.82% 2.56% 46.15% Count of Visits 60 25 5 90/ 1 Data Yes CondProb 66.67% 27.78% 5.56% 100.00% Data Yes CondProb 0.6666666667 0.2777777778 0.0555555556 1 JointProb 0.3076923077 0.1282051282 0.0256410256 0.4615384615 Count of Visits Count of Visits Enclosed Mall Visits Often Occasional Never Grand Total No CondProb Count of Visits 60 25 5 90 Enclosed Mall Visits Often Occasional Never Grand Total JointProb Count of Visits 60 25 5 90 JointProb 20 35 50 105 4/ 21 1/ 3 10/ 21 1/ 1 4/ 39 7/ 39 10/ 39 7/ 13 20 35 50 105/ 1 No CondProb 19.05% 33.33% 47.62% 100.00% JointProb 10.26% 17.95% 25.64% 53.85% No CondProb 0.1904761905 0.3333333333 0.4761904762 1 JointProb 0.1025641026 0.1794871795 0.2564102564 0.5384615385 20 35 50 105 done with PivotTable and Fractional Number Format Done with formulas and % format Done with PivotTable and Custom Fraction Number Format 25 a) b) P(Checking) P(Savings) P(C & S) P(C OR S) = 0.8 + 0.6 - 0.5 = P(Not C OR S) = 1 - +0.9 = 0.8 0.6 0.5 0.9 0.1 26 P(1st truck available) = P(2nd truck available) = P(Both 1st and 2nd truck available) = Step 1: Find P(1st OR 2nd) = 0.75 + 0.5 - 0.3 = Step 2: Find P(neither available) = 1 - P(1st OR 2nd) = 1 - 0.95 = 0.75 0.5 0.3 Remember: P(1st OR 2nd) means 1st is working, or second is 0.95 working or Both are working. 0.05 So that means that 1 - P(1st OR 2nd) means neither are working Defective Toothbrushes = Non-Defective Toothbrushes = Total a) b) P(first 2 brushes are not defective) =3/20*2/19 = 0.0157894736842105 P(first 2 brushes are not defective) =17/20*16/19 = 0.715789473684211 3 17 20 0.0157894737 Sample Space Changes 0.7157894737 Sample Space Changes c7466754d66532ff4fbbea246d745bff7605e657.xls - Ex29 Potential for advancement Fair Sales Ability Below Ave. Ave. Above Ave. Totals Good 16 45 93 154 Excellent Totals 12 60 72 144 22 50 45 150 135 300 202 500 Sales Ability Fair Below Ave. Ave. Above Ave. Joint Probabilities Good 0.032 0.09 0.186 Excellent 0.024 0.12 0.144 0.044 0.09 0.27 SUM = a) b) c) Contingency Table: Good for Nominal Data P(Above Ave. Sales Ability AND Excellent Potential for advancement) = 135/500 = 0.27 0.27 1 c7466754d66532ff4fbbea246d745bff7605e657.xls - Ex29 (2) Potential for advancement Good Sales Ability 16 45 93 154 Excellent 12 60 72 144 Totals 22 50 45 150 135 300 202 500 Sales Ability Below Ave. Ave. Above Ave. Fair Joint Probabilities Good 3.20% 9.00% 18.60% Excellent 2.40% 12.00% 14.40% 4.40% 9.00% 27.00% SUM = Contingency Table: Good for Nominal Data P(Above Ave. Sales Ability AND Excellent Potential for advancement) = 135/500 = a) b) c) PotForAd Sales Ability Below Ave. Ave. Above Ave. Grand Total Count 16 45 93 154 0.27 0.27 Data Fair CondProb 32.00% 30.00% 31.00% 30.80% JointProb 3.20% 9.00% 18.60% 30.80% Count 12 60 72 144 Good CondProb 24.00% 40.00% 24.00% 28.80% JointProb 2.40% 12.00% 14.40% 28.80% Count 22 45 135 202 Excellent CondProb 44.00% 30.00% 45.00% 40.40% JointProb 4.40% 9.00% 27.00% 40.40% 100.00% 1 2 3 4 5 6 Sales Ability Below Ave. Below Ave. Below Ave. Below Ave. Below Ave. Below Ave. Potential for advancement Fair Fair Fair Fair Fair Fair 7 Fair Below Ave. Ave. Above Ave. Totals Below Ave. Fair 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 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Above Ave. Above Ave. Above Ave. Above Ave. Above Ave. Above Ave. Above Ave. Above Ave. Above Ave. Fair Fair Fair Fair Fair Fair Fair Fair Fair Good Good Good Good Good Good Good Good Good Good Good Good Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Fair Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good 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Problems not assigned ==> P(A1|B) = P(A1)*P(B|A1) P(A1)*P(B|A1) + P(A2)*P(B|A2) Prior Probability = The initial Probability based on the Event Ai (These events are mutually Exclusive and Collectively present Level of Information Prior Probability = P(Ai) = Exhaustive) P(Ai)*P(B|Ai) P(Ai|B) = data in white cells is given P(A1)*P(B|A1) + P(A2)*P(B|A2) + … + P(An)*P(B|An) Conditional Probability = Probability that B given that Ai has already happened = P(B|Ai) = Has Disease in Pop. = A1 0.05 Not Have Disease in Pop. = A2 0.95 1 P(B) = Joint Probability = Both are true = P(Ai and B) = Posterior Probability = A revised Probability based on Additional Information = P(Ai| B) = 0.9 0.045 0.24 0.15 0.1425 0.1875 0.76 1 P(A1|B) = P(Ai|B) = P(A1)*P(B|A1) P(Ai)*P(B|Ai) P(A1)*P(B|A1) + P(A2)*P(B|A2) B1 = P(A1)*P(B|A1) + P(A2)*P(B|A2) + … + P(An)*P(B|An) Defective B2 = Key Concept!!!!! Particular Joint Probability/SUM(of joint Probabilities) Not Defective Event Ai (These events are mutually Exclusive and Collectively Exhaustive) Prior Probability = The initial Probability based on the present Level of Information Prior Probability = P(Ai) = Conditional Probability = Probability that B1 (defective) given that Ai has already happened = P(B|Ai) = Posterior Probability = A revised Probability based on Additional Information = P(Ai|B1) = Joint Probability = Both are true = P(Ai and B) = Prob. Chips bought from Hall = A1 0.3 0.03 0.009 0.2307692308 Prob. Chips bought from SS = A2 0.2 0.05 0.01 0.2564102564 Prob. Chips bought from CC = A3 0.5 1 P(B) = 0.04 0.02 0.039 0.5128205128 1 Event Ai (These events are mutually Exclusive and Collectively Exhaustive) Prior Probability = The initial Probability based on the present Level of Information Prior Probability = P(Ai) = Conditional Probability = Probability that B (notdefective) given that Ai has already happened = P(B2|Ai) = Posterior Probability = A revised Probability based on Additional Information = P(Ai|B2) = Joint Probability = Both are true = P(Ai and B) = Prob. Chips bought from Hall = A1 0.3 0.97 0.291 0.3028095734 Prob. Chips bought from SS = A2 0.2 0.95 0.19 0.197710718 Prob. Chips bought from CC = A3 0.5 1 P(B) = 0.96 0.48 0.961 0.4994797086 1 Total 1 data in white cells is given Given Information: Prior Probabilities = P(Ai) Probability that part came from Hall = Probability that part came from SS = Probability that part came from CC = = P(A1) P(A2) P(A3) 0.3 0.2 0.5 1 Additional Information: Part is Defective = Part is Not Defective = = Given that part came from Hall, probability that part is defective = Given that part came from SS, probability that part is defective = Given that part came from CC, probability that part is defective = Given that part came from Hall, Probability that part is not defective = Given that part came from SS, Probability that part is not defective = Given that part came from CC, probability that part is not defective = B1 B2 Conditional Probability = P(Bi) 0.03 0.05 0.04 0.97 0.95 0.96 P(B1|A1) P(B1|A2) P(B1|A3) P(B2|A1) P(B2|A2) P(B2|A3) 1 1 1 Joint Probabilities Probability that part came from Hall AND it is defective = P(A1) * P(B1|A1) = 0.3 * 0.03 = Probability that part came from SS AND it is defective = P(A2) * P(B1|A2) = 0.2 * 0.05 = Probability that part came from CC AND it is defective = P(A3) * P(B1|A3) = 0.5 * 0.04 = 0.009 0.01 0.02 0.039 Probability that part came from Hall AND it is NOT defective = P(A1) * P(B2|A1) = 0.3 * 0.97 = Probability that part came from SS AND it is NOT defective = P(A2) * P(B2|A2) = 0.2 * 0.95 = Probability that part came from CC AND it is NOT defective = P(A3) * P(B2|A3) = 0.5 * 0.96 = 0.291 0.19 0.48 0.961 0.009 0.010 0.020 0.039 Joint Probabilities Because theses are all the possibilities 0.291 0.190 0.480 0.961 1 Array Formula Posterior Prob. Posterior Probabilities = Revised Probabilities based on additional Information Posterior Probabilities (Revised Probability Based On Additional Information) = Probability that chip is from Hall, given that we have selected a defective chip = 0.009/0.039 = 9/39 = Posterior Probabilities (Revised Probability Based On Additional Information) = Probability that chip is from SS, given that we selected a defective chip = 0.01/0.039 = 10/39 = Posterior Probabilities (Revised Probability Based On Additional Information) = Probability that chip is from CC, given that we selected a defective chip = 0.02/0.039 = 20/39 = Posterior Probabilities (Revised Probability Based On Additional Information) = Probability that chip is from Hall, given that we have selected a non-defective chip = 0.291/0.961 = 291/961 = Posterior Probabilities (Revised Probability Based On Additional Information) = Probability that chip is from SS, given that we selected a non-defective chip = 0.19/0.961 = 190/961 = Posterior Probabilities (Revised Probability Based On Additional Information) = Probability that chip is from CC, given that we selected a non-defective chip = 0.48/0.961 = 480/961 = check P(A1|B1) = P(A1|B1) 0.2307692308 0.2307692308 0.2307692308 0.2307692308 P(A2|B1) 0.2564102564 0.2564102564 0.2564102564 0.2564102564 P(A3|B1) check 0.5128205128 1 0.5128205128 1 0.5128205128 0.5128205128 P(A1|B2) 0.3028095734 0.3028095734 0.3028095734 0.3028095734 0.197710718 0.197710718 0.197710718 0.197710718 0.4994797086 1 0.4994797086 1 0.4994797086 0.4994797086 P(A2|B2) P(A3|B2) = (P(A3)*P(B2| A3))/(Sum of all Joint Probabilities) check Prior Probabilities Probability that Game Played at Night Probability that Game Played in Day N D P(N) P(D) 0.7 0.3 1 Conditional Probabilities Win Given at Night Win Given at Day Loss Given at Night Loss Given at Day W|N W|D L|N L|D P(W|N) P(W|D) P(L|N) P(L|D) 0.5 0.9 0.5 0.1 Joint Probabilities Probability that Game Played at Night * Win Given at Night = P(N) * P(W|N) = 0.7 * 0.5 Probability that Game Played in Day * Win Given at Day = P(D) * P(W|D) = 0.3 * 0.9 Probability that Game Played at Night * Loss Given at Night = P(N) * P(L|N) = 0.7 * 0.5 Probability that Game Played in Day * Loss Given at Day = P(D) * P(L|D) = 0.3 * 0.1 check 1 1 Posterior Probability = Revised Probability Based on Additional Information = Particular Joint/Sum(all joint) 0.35 P(W|N) 0.564516129 P(W|D) 0.435483871 0.35 P(L|N) 0.9210526316 0.03 0.38 P(L|D) 0.0789473684 0.27 0.62 check 1 1 F NF P|F P|NF Prior Probabilities Probability that student finish assignments = Probability that student not finish assignments = Conditional Probabilities Probability they pass class, given that they finished assignments = Probability they pass class, given that they not finished assignments = P(F) P(NF) 0.8 0.2 P(P|F) 0.9 P(P|NF) 0.6 Joint Probabilities P(F) * P(P|F) = P(NF) * P(P|NF) = 0.72 = 0.12 = 0.84 = Posterior Probabilities Probability that they finished assignments given that they passed the class = P(F|P) Probability that they not finish assignments, Given that they passed the class = P(NF|P) 0.8571428571 0.857143 0.1428571429 0.142857 1 0.72 0.12 0.84 Prior Probabilities Cash C Credit Card CC Debt Card DC >$50.00 P(C) P(CC) P(DC) Sales paid for with Cash Sales paid for with Credit Card Sales paid for with Debt Card 50 Purchanse was for more than $50.00 Probability that Purchanse was for more than $50.00, Given that Sales P(>$50.00|C) paid for with Cash Probability that Purchanse was for more than $50.00, Given that Sales P(>$50.00|CC) paid for with Credit Card Probability that Purchanse was for more than $50.00, Given that Sales P(>$50.00|DC) paid for with Debt Card P(C) * P(>$50.00|C) P(CC) * P(>$50.00|CC) P(DC) * P(>$50.00|DC) P(C|>$50.00) Probability that Cash was paid Given that sales was >$50.00 P(CC|>$50.00) Probability that Credit Card was paid Given that sales was >$50.00 P(DC|>$50.00) Probability that Debt Card was paid Given that sales was >$50.00 0.3 0.3 0.4 1 Conditional Probabilities 0.2 0.9 0.6 Joint Probabilities 0.06 0.27 0.24 0.57 Posterior Probabilities 0.1052631579 0.105263 0.4736842105 0.473684 0.4210526316 0.421053 1 1 Prior Leave garage open Not Leave garage open Conditional Stuff Stolen Given, Leave garage open Stuff Stolen Given, Not Leave garage open Doors open given Stolen 0.25 0.75 0.05 0.01 0.625 Gen der Going To College Root or Prior Probabilities P(F) P(M) P(WC|F) P(NotWC|F) P(WC|M) P(NotWC|M) c 0.8 Female 0.2 Male Conditional Probabilities 0.9 Went To College 0.1 Did Not Go To College 0.78 Went To College 0.22 Did Not Go To College a P( select a female from whole pool who is female AND did not go to college) b Independence means that one event does not have an affect on another event. The Sample Space is not changed when the second event occurs. P(NotWC|F) = P(F) 0.08 Root Prob Conditional Prob F M Went To College WC NotWC WC NotWC P(F) P(M) P(WC|F) P(NotWC|F) Joint Prob 0.9 P(F) * P(WC|F) = 0.8 * 0.9 0.1 P(F) * P(NotWC|F) = 0.8 * 0.1 0.72 0.08 0.78 P(M) * P(WC|M) = 0.2 * 0.78 0.22 P(M) * P(NotWC|M) = 0.2 * 0.22 0.156 0.044 0.8 0.2 Did Not Go To College P(WC|M) P(NotWC|M) c 1 It equals 1 because all the possible probabilities are included No, because if you know gender, it changes the probability that they go to college. Variable 0.9 w 0.1 Nw 4 F1 F2 F3 F4 Flight arrives within 15 minutes of scheduled arrival Flight does not arrive within 15 minutes of scheduled arrival We select this # of flights to study: Flight 1 Flight 2 Flight 3 Flight 4 If you assume independence, then P(F1 w 15 AND F2 w 15 AND F3 w 15 AND F4 w 15) = If you assume independence, then P(F1 Nw 15 AND F2 Nw 15 AND F3 Nw 15 AND F4 Nw 15) = a b c Possible Outcomes Sample space = (# of outcomes)^(# of Trials) = 2^4 = Flights and w or Nw 1 w,w,w,w 2 Nw,w,w,w 3 w,Nw,w,w 4 w,w,Nw,w 5 w,w,w,Nw 6 Nw,Nw,w,w 7 w,Nw,Nw,w 8 w,w,Nw,Nw 9 Nw,w,w,Nw 10 Nw,w,Nw,w 11 w,Nw,w,Nw 12 Nw,Nw,Nw,w 13 w,Nw,Nw,Nw 14 Nw,w,Nw,Nw 15 Nw,Nw,w,Nw 16 Nw,Nw,Nw,Nw 0.6561 0.00010000 16 # of Nw 0 0.6561 1 0.0729 1 0.0729 1 0.0729 1 0.0729 2 0.0081 2 0.0081 2 0.0081 2 0.0081 2 0.0081 2 0.0081 3 0.0009 3 0.0009 3 0.0009 3 0.0009 4 0.00010000 1 chapter 6: Binomial? Fixed # of trials? Yes, n = 4 Each trial independent? Yes, n = 4 S/F each time Yes, n = 4 Probability stays the same Yes pi = .9 each trial? 0.6561 0.2916 0.0486 0.0036 0.0001 0 Random Variable X = # of Nw = # of flights not within 15 minutes 0 1 2 3 4 1 P(X = 0) = 4 P(X = 1) = 6 P(X = 2) = 4 P(X = 3) = 1 P(X = 4) = 16 P(X >= 1) = P(X >= 1) = 0.6561 0.2916 0.0486 0.0036 0.0001 check check check check check 0.3439 0.3439 cumulative 0 = exact value, 1 = up to that value 0 0 0 0 0 >=1 1 0.3439 1 Production workers Supervisors Secretaries President a b c d 57 P(Prod) 40 P(Supe) 2 P(Secr) 1 P(Pres) 100 P(Prod) P(Prod OR Supe) Yes P(NOTProd OR NOTSupe) P(NOTProd OR NOTSupe) 0.57 0.97 0.03 0.03 0.57 0.4 0.02 0.01 1 X = Random Variable = # of hits in 4 tries a b c P(X) 4 P(4) 0 P(0) 1 P(1) Probability that missle hits its target Probability that missle NOT hit its target # of Outcomes = # of Trials = # of Missles sent Sample Space Size = # of possible outcomes in sample space = Outcomes^#Trials = P(All 4 Hit) = P(None Hit) = P(at least 1 hit) = HT NHT P(HT) P(NHT) 2 4 16 0.4096 0.0016 0.9984 0.8 0.2 ch 6 check check 0.4096 0.0016 0.9984 Cumulative: 0 = exact value, 1 = up to that value 0 0 1 # of Students Graduating = # of students who are graduating and going to college Students picked at random to carry flags at graduation = 90 50 2 check Probability that the student 1 selected to carry the flag is going to college = 50/90 = 0.5555556 Root or Prior Probability Probability that the student 2 selected to carry the flag is going to college = 49/89 = 0.5505618 Conditional Prob. Probability that both students selected to carry flag are going a to college 0.3058677 Joint Probability Probability that one of the 2 students selected to carry flag b are going to college 0.4993758 0.5555555556 0.5505617978 selection 1 selection 2 P(X) Possibility 1 Going to college Not going to college 0.24968789 50/90*40/89 Possibility 2 Not going to college Going to college 0.24968789 40/90*50/89 0.49937578 Probability that a 60 year old man will survive 1 year Probability that a 60 year old man will Not survive 1 year # of Outcomes = Policy offered to this many men = Trials = Sample Space = # of Possible Outcomes = a Probability that all 5 survive? Probability that none survive? b Probability that at least one does not survive? 60S1Y 60NOTS1Y P(60S1Y) P(60NOTS1Y) 0.98 0.02 chapter 6: Fixed # Trials? Each trial independent? Probability stay same each trial? Probability of Success the same each trial? 2 5 32 0.9039207968 0.00000000320 0.0960792032 check 0.9039207968 0.00000000320 0.0960792032000 0.0960792032000 Not Survive P(That 0 Not survive) P(That 1 Not survive) P(That 2 Not survive) P(That 3 Not survive) P(That 4 Not survive) P(That 5 Not survive) Yes n = Yes Yes Yes 0 1 2 3 4 5 5 0.9039207968000 0.0922368160000 0.0037647680000 0.0000768320000 0.0000007840000 0.0000000032000 cumulative 0 0 0 0 0 0 0.0960792032 Survive P(That 0 Survive) P(That 1 Survive) P(That 2 Survive) P(That 3 Survive) P(That 4 Survive) P(That 5 Survive) 0 1 2 3 4 5 0.0000000032000 0.0000007840000 0.0000768320000 0.0037647680000 0.0922368160000 0.9039207968000 cumulative 0 0 0 0 0 0 Probability that Homes contain Security System Probability that Home does not contain Security System # of outcomes = # of Trials = Homes selected at random = Sample Space = # of possible outcomes = 62 a b c d P(3) P(0) P(1 or 2 or 3) = 1 - P(0) = Independent 0.4 success = home has SS = Not success = home Not 0.6 have SS = 2 3 8 0.064 0.216 0.784 Probability that Homes contain Security System Probability that Home does not contain Security System # of outcomes = # of Trials = Homes selected at random = Sample Space = # of possible outcomes = # of Homes # of homes with SS Probability that Homes contain Security System Probability that Home does not contain Security System # of outcomes = # of Trials = Homes selected at random = 63 a P(3) Pr(3 straight homes selected with SS) = b P(0) Pr(3 straight homes selected with No SS) = c P(X>=1) P(at least 1 has a SS) = P(1 or 2 or 3) = d 0.4 success = home has SS = 0.6 Not success = home Not have SS = 2 3 8 10 4 0.4 success = home has SS 0.6 2 3 0.0333333 0.1666667 0.8333333 0.8333333333 s ns check 3 successes 0 0.4 Root or Prior 1 0.3333333333 Conditional Probability 2 0.25 Conditional Probability 0.0333333333 product check 2 successes 0 0.4 s 1 0.3333333333 s 2 0.75 ns 0.1 product check 1 successes 0 0.4 s 1 0.6666666667 ns 2 0.625 ns 0.1666666667 product check 0 successes 0 0.6 Root or Prior 1 0.5555555556 Conditional Probability 2 0.5 Conditional Probability 0.1666666667 product # of Possible # of Trials = Homes selected at random = # of 1 2 3 successes Outcomes 1s s s 3 2s s ns 2 3s ns s 2 4 ns s s 2 5s ns ns 1 6 ns s ns 1 7 ns ns s 1 8 ns ns ns 0 Prob of occurance 0.03333333 0.1 0.1 0.1 0.16666667 0.16666667 0.16666667 0.16666667 1 # Families that did there own taxes # Families that had pro do there taxes # Families that had H&R Block do there taxes Total # Families = a b c d P(OT) P(OT) P(Selecting 2nd OT | that you already selected a OT on 1st try) P(select one OT AND then a second OT) = P(OT) P(Selecting 2nd OT | that you already selected a OT on 1st try) P(Selecting 3rd OT | that you already selected a OT on first 2 tries) P(select one OT AND then a second OT) = P(select 2 families that did not have HR Block do Taxes) 10 OT 7 PDT 3 HRDT 20 0.5 0.5 Root 0.4736842105 Conditional 0.2368421053 Joint 0.5 Root 0.4736842105 Conditional 0.4444444444 Conditional 0.1052631579 Joint 0.7157894737 P(OT) P(PDT) P(HRDT) 0.5 0.35 0.15 1 total members W women M men 12 3 P(W) 9 P(M) # of outcomes = count s and count Ns = # of committee members to select = # of Trials = # of times that you choose and get either a man or a woman # of possible Outcomes in Sample Space = (# of Outcomes)^(# of trials) a b P(3 straight men selected for committee) P(at least one woman) = P(1W OR 2W OR 3W) = 1 - P(0W) = 1 - P(3 straight men selected for committee) = # of possible Outcomes Prob on first selection only 0.25 Success = Choose woman 0.75 not success = choose man 1 s Ns use hypergeopmetric whne sample space changes 2 chapter 6: Hypergeometric 3 X 8 P(X) 0.3818182 Conditional Probability (Saample space changes) 0 0.00454545 1 0.12272727 2 0.49090909 0.6181818 0.6181818 3 0.38181818 # of possible Outcomes in Sample Space = (# of Outcomes)^(# of trials) # of Women selected 1s s s 2 Ns s s 3s Ns s 4s s Ns 5 Ns Ns s 6 Ns s Ns 7s Ns Ns 8 Ns Ns Ns Probability for Occurance of Given Outcome 3 0.0045454545 <++ formula 2 0.0409090909 <++ formula 2 0.0409090909 2 0.0409090909 1 0.1636363636 <++ formula 1 0.1636363636 1 0.1636363636 0 0.3818181818 <++ formula 1 Shareholders made $ Shareholders lost $ Total a b c d P( Company from list randomly selected, CEO paid > $1 M ) P( Company from list randomly selected, CEO paid > $1 M OR Shareholders lost $ ) P( Company from list randomly selected, CEO paid > $1 M Given That Shareholders lost $ ) P( 2 Companies from list randomly selected, and both had CEO paid > $1 M) CEO paid > $1 M CEO paid < $1 M Total 2 11 4 3 6 14 0.3 0.45 0.5714285714 0.0789473684 13 7 20 Proportion of company's potential market that read the Gazette = Proportion of company's potential market that don't read the Gazette = Proportion of Gazette Readers who remember the company's ad = Proportion of Gazette Readers who DO NOT remember the company's ad = a b P(that Company's potential market reads and remembers the ad) P(that Company's potential market reads and DOES NOT remembers the ad) 0.6 0.4 1 0.85 0.15 1 because you start with a group of readers, and then since 85% of all readers remember the company's ad, that means that 85% of this group will remember the 0.51 ad. 0.09 # of Presidents # of Vice-Presidents Company President has season tickets to Baseball Game Company President always takes 1 of 4 vice presidents President claims that he picks Vice-Presidents at random Success = Being selected Not Success = Not Being Selected # of times not selected What is probability of not getting selected 5 straight times? 1 4 s Ns 5 0.2373 P(Selecting 1 Vice President) P(Selecting of not getting selected) 0.25 0.75 1 Formatted CDs that were perfect Formatted CDs that were usable but had bad sectors Formatted CDs that were NOT usable Total CDs a b P(CD not Perfect) = P(CD Not Usable)) = 857 P 112 UBBS 31 NU 1000 P(P) P(UBBS) P(NU) 0.143 0.143 0.2167832 Conditional Probability 0.857 0.112 0.031 1 P(Bank Stock Increase) = P(Utility Stock Increase) = P(Bank Stock Not Increase) = P(Utility Stock Not Increase) = Outcome 1 Outcome 2 Outcomes = Total Trials = 2 stocks Sample Space = a b c 0.7 0.6 0.3 0.4 Up Down 2 2 4 P(Both Increase) P(Bank Up and Utility Down) P(at least one stock goes up) = P(B Up AND U Up OR B Up AND U Down OR B Down AND U Up) Possible Outcomes s Ns 0.42 Assuming Independence 0.28 Assuming Independence 0.88 Bank S 1 Up 2 Up 3 Down 4 Down X Utility S Up Down Up Down P(X) 0 P(0) 1 P(1) 2 P(2) # of Successes 2 1 1 0 P(X) 0.12 0.46 0.42 1 0.42 0.28 0.18 0.12 1 Marketing Research Company provides assessments of how well women stores will do in mall From Past Data: Root Prob Conditional Prob Store Do Good Good P(G) 0.60 Earn Profit given that Store Do Good Earn Profit Good P(G) 0.60 Not Earn Profit given that Store Do Good Not Earn Profit Store Do Fair Fair P(F) 0.30 Earn Profit given that Store Do Fair Earn Profit Fair P(F) 0.30 Not Earn Profit given that Store Do Fair Not Earn Profit Store Do Poor Poor P(P) 0.10 Earn Profit given that Store Do Poor Earn Profit Poor P(P) 0.10 Not Earn Profit given that Store Do Poor Not Earn Profit 1.00 check Store Do Good Store Do Fair Store Do Poor Earn Profit given that Store Do Good Earn Profit given that Store Do Fair Earn Profit given that Store Do Poor 0.60 0.30 0.10 0.8 0.6 0.2 P(Earn Profit | G) P(Not Earn Profit | G) P(Earn Profit | F) P(Not Earn Profit | F) P(Earn Profit | P) P(Not Earn Profit | P) Joint Prob 0.8 P(Good AND Earn Profit) 0.2 P(Good AND Not Earn Profit) 0.6 P(Fair AND Earn Profit) 0.4 P(Fair AND Not Earn Profit) 0.2 P(Poor AND Earn Profit) 0.8 P(Poor AND Not Earn Profit) 1 1 1 P(Poor | Earned a Profit) 0.0294117647 0.0294117647 0.48 0.12 0.18 0.12 0.02 0.08 1 B1 B1 MP ST Box 1 Mesh Polo Shirts Box 1 Super-T Shirts B2 B2 MP ST Box 2 Mesh Polo Shirts Box 2 Super-T Shirts 25 15 40 30 10 40 Joint Probabilities Joint Probabilities Joint Probabilities Joint Probabilities 2 0.5 0.5 0.625 0.375 0.750 0.25 80 55 25 0.6875 0.3125 Box selecetd at random, then shirt selected at random, it was a MP. What is P(B1 | MP) Rott Probabilities Rott Probabilities Conditional Probabilities Conditional Probabilities Conditional Probabilities Conditional Probabilities # of B1 # of B2 Total Boxes = P(B1) P(B2) P(MP|B1) P(ST|B1) P(MP|B2) P(ST|B2) Total Shirts Total MP Shirts Total ST Shirts P(MP from whole pile of Shirts) P(ST from whole pile of Shirts) 0.625 P(B1) * P(MP|B1) P(B1) * P(ST|B1) P(B2) * P(MP|B2) P(B2) * P(ST|B2) 1 1 0.3125 0.1875 0.3750 0.1250 checks 1 1 1 80 1 0.625 Posterior Prob 0.50000 Posterior Prob Posterior Prob 0.5 Posterior Prob 1 P(B1 | MP) P(B2 | MP) P(B1 | ST) P(B2 | ST) 0.4545 0.5455 0.6000 0.4000 1.0000 1.0000 B1 MP Box 1 Mesh Polo Shirts B1 ST Box 1 Super-T Shirts B2 MP Box 2 Mesh Polo Shirts B2 ST Box 2 Super-T Shirts 25 15 40 30 10 40 Conditional Root P(MP | B1) P(ST | B1) P(B1) 0.5000 P(ST | B2) Joint 0.6250 P(B1) * P(MP | B1) 0.3750 P(B1) * P(ST | B1) 0.5000 P(B2) Joint for Posterior need to be grouped by First part of Conditional Proba 0.3125 0.1875 P(B1) * P(MP | B1) P(B2) * P(MP | B2) 0.7500 P(B2) * P(MP | B2) 0.2500 P(B2) * P(ST | B2) 0.3750 0.1250 0.5000 P(B1) * P(ST | B1) P(B2) * P(ST | B2) 1 0.6875 0.1875 0.1250 P(ST from whole 0.3125 pile of Shirts) 1 0.6 0.4 0.3125 0.3750 P(MP from whole 0.6875 pile of Shirts) 0.5000 P(MP | B2) check Posterior Probability (Revised Prob given new information) - the new information here is Mens Polo and now we go backward to find prob that it's from box 1 0.4545454545 0.5454545455 0.3125 SD Odds of free soft drink Odds of Not winning free soft drink P Odds of not winning free pizza Odds of Not winning free pizza Outcomes Trials = Sample space = Outcomes ^ Trials P(Win P) = P(Win SD) = a b c d P(Win SD or P) P(Not Win) P(Not Win in 3 tries) P(at leat one prize on next three visits) P(SD) P(NOT WIN SD) P(P) P(NOT WIN P) 0.1 10 10 100 0.1000 0.1 TRUE 50 2 12 100 0.0200 0.02 TRUE 0.9 0.02 0.98 2 Outcomes = win or not win 1 2 prob 0.1 0.02 AND Or AND P((Win P AND NotWin SD)) OR 0.1160 =1/50*9/10+1/10*49/50 P(P)*P(NotSD)+P(S)*P(NotP) (Win SD AND NotWin P)) 0.8840 0.6908 3 0.3092 Possible outcomes 1 P(W) 2 P(Not W) 0.1160 0.8840 0.1200 All Possible = Subgroup size = Select 3 numbers from 10 possible n r 10 3 In this lottery, if a number is selected for one position in sub group, it cannot be used for another position in the subgroup - so 337 could not be used because there are 2 3s, but 307 or 370 could be used. Combination (2,1 is not different from 1,2) = n!/(r!*(n-r)!) Permutation (2,1 is different from 1,2) = n!/((n-r)! a b c Permutations = Permutations = P(win) = # tickets bought = P(win when you buy 3 tickets) = P(not win after buying 3 ticket) = 720 720 0.0013888889 3 0.0041666667 0.9958333333 Ways to make hamburger Items avaiable on Hamberger: Mustard Ketchup Onion pickel tomato relish mayonnaise lettuce Count of items: You may choose to have, or ommit, any combination of items, but Onion,Relish is the same as Relish, Onion. Thus use Combination. Order does not matter 256 8 =n Combinations r 0 items on Burger r0 1 items on Burger r1 2 items on Burger r2 3 items on Burger r3 4 items on Burger r4 5 items on Burger r5 6 items on Burger r6 7 items on Burger r7 8 items on Burger r8 Total The advertisement is correct because if you find the combination for each possible number from 0 (no items on burger) to 8 (all items on burger), and then add them up you get 256. Testbook: Outcomes = Trials = n Sample Space = (# of Outcomes)^(#of trials) = 2 8 256 0 1 2 3 4 5 6 7 8 1 8 28 56 70 56 28 8 1 256 Beijing Xi'an B X P(B) P(X) P(B AND X) P(at least 1 site) = P(B or X or Both) = P(B OR X) P(B) = 0.6 P(B AND X) = 0.3 0.4 P(X) = 0.6 P(B) = 0.6 0.4 P(X) = 0.4 0.3 P(B AND X) = 0.3 0.7 P(B AND X) = 0.3 P(X) = 0.4 Stop Smoking Gum S NS # in Group of smokers that try to quit = # of Trials = n = # of Outcomes = Stop or Not Stop = Total Possible Outcomes = P(0 stop smoking) = P(at least 1 stop smoking) = P(1 stop OR 2 stop OR 3 stop OR 4 stop) = S w SSG NS w SSG P(S w SSG) P(NS w SSG) 0.6 0.4 4 2 16 0.0256 0.9744 check 0.9744 # of people Possible outcomes # in Group of smokers that try to quit = # of Trials = n = Probability that STOP 1st person 2nd person 3rd person 4th person 1S 2 NS 3S 4S 5S 6 NS 7 NS 8 NS 9S 10 S 11 S 12 S 13 NS 14 NS 15 NS 16 NS S S NS S S NS S S NS NS S NS S NS NS NS S S S NS S S NS S NS S NS NS NS S NS NS S S S S NS S S NS S NS NS NS NS NS S NS 4 3 3 3 3 2 2 2 2 2 2 1 1 1 1 0 0.1296 0.0864 0.0864 0.0864 0.0864 0.0576 0.0576 0.0576 0.0576 0.0576 0.0576 0.0384 0.0384 0.0384 0.0384 0.0256 1 # of Exterior possibilities # of Interior possibilities # of different ways to offer Exterior and Interior plans to Customers = 5 3 15 Breaks Do Not work Breaks Do work Steering Does Not Work Steering Does Work Assume events are Independent B BW S SW P(B) P(BW) P(S) P(SW) If 1 or the other problem is present (but not both) car is called a: Lemon P(B OR S) = P(B)*P(SW)+P(S)*P(BW) If Both Problems are present Car Hazard P(B AND S) = P(B) * P(S) is called a: 0.15 0.85 1 0.05 0.95 1 You must do it this way because there are 2 possibilities that must be added, but each possibility is a "AND"-joint-multiplying situation. The first possibility is that the breaks don't work AND the Steering works (the AND implies multiplying) OR the steering does not work 0.185 AND the Breaks do work (the AND implies multiplying). 0.0075 how ma ny license pla tes a re possible? # of pla ces tha t License Pla te numbers a re a llowed: # of pla ces tha t License Pla te letters a re a llowed: Number of Different Numbers Number of Different Letters # of license pla tes 3 3 10 26 17,576 ,000 = 10*10*10*26 *26 *26 = 10^3*26 ^3 10 PO SSIBLE CHARACTERS HERE = = > 10 PO SSIBLE CHARACTERS HERE = = > 10 PO SSIBLE CHARACTERS HERE = = > 0 26 PO SSIBLE CHARACTERS HERE = = > 000 AAA 001 AAA 002 AAA 003 AAA 004 AAA 005 AAA 006 AAA 007 AAA 008 AAA 009 AAA 010 AAA 011 AAA 012 AAA 013 AAA 014 AAA 015 AAA 016 AAA 017 AAA 018 AAA 019 AAA 020 AAA 021 AAA 022 AAA 023 AAA 024 AAA 025 AAA 026 AAA 027 AAA 028 AAA 029 AAA 030 AAA 031 AAA 032 AAA 033 AAA 034 AAA 035 AAA 036 AAA 037 AAA 038 AAA 039 AAA 04 0 AAA 04 1 AAA 04 2 AAA 04 3 AAA 04 4 AAA 04 5 AAA 04 6 AAA 04 7 AAA 04 8 AAA 04 9 AAA 050 AAA 051 AAA 052 AAA 053 AAA 054 AAA 055 AAA 056 AAA 057 AAA 058 AAA 059 AAA 06 0 AAA 06 1 AAA 06 2 AAA 06 3 AAA 06 4 AAA 06 5 AAA 06 6 AAA 06 7 AAA 06 8 AAA 06 9 AAA 070 AAA 071 AAA 072 AAA 073 AAA 074 AAA 075 AAA 076 AAA 077 AAA 078 AAA 079 AAA 080 AAA 081 AAA 082 AAA 083 AAA 084 AAA 085 AAA 086 AAA 087 AAA 088 AAA 089 AAA 09 0 AAA 09 1 AAA 09 2 AAA 09 3 AAA 09 4 AAA 09 5 AAA 09 6 AAA 09 7 AAA 09 8 AAA 09 9 AAA 100 AAA 101 AAA 102 AAA 103 AAA 104 AAA 105 AAA 106 AAA 107 AAA 108 AAA 109 AAA 110 AAA 111 AAA 112 AAA 113 AAA 114 AAA 115 AAA 116 AAA 117 AAA 118 AAA 119 AAA 120 AAA 121 AAA 122 AAA 123 AAA 124 AAA 125 AAA 126 AAA 127 AAA 128 AAA 129 AAA 130 AAA 131 AAA 132 AAA 133 AAA 134 AAA 135 AAA 136 AAA 137 AAA 138 AAA 139 AAA 14 0 AAA 14 1 AAA 14 2 AAA 14 3 AAA 14 4 AAA 14 5 AAA 14 6 AAA 14 7 AAA 14 8 AAA 14 9 AAA 150 AAA 151 AAA 152 AAA 153 AAA 154 AAA 155 AAA 156 AAA 157 AAA 158 AAA 159 AAA 16 0 AAA 16 1 AAA 16 2 AAA 16 3 AAA 16 4 AAA 16 5 AAA 16 6 AAA 16 7 AAA 16 8 AAA 16 9 AAA 170 AAA 171 AAA 172 AAA 173 AAA 174 AAA 175 AAA 176 AAA 177 AAA 178 AAA 179 AAA 180 AAA 181 AAA 182 AAA 183 AAA 184 AAA 185 AAA 186 AAA 187 AAA 188 AAA 189 AAA 19 0 AAA 19 1 AAA 19 2 AAA 19 3 AAA 19 4 AAA 19 5 AAA 19 6 AAA 19 7 AAA 19 8 AAA 19 9 AAA 200 AAA 201 AAA 202 AAA 203 AAA 204 AAA 205 AAA 206 AAA 207 AAA 208 AAA 209 AAA 210 AAA 211 AAA 212 AAA 213 AAA 214 AAA 215 AAA 216 AAA 217 AAA 218 AAA 219 AAA 220 AAA 221 AAA 222 AAA 223 AAA 224 AAA 225 AAA 226 AAA 227 AAA 228 AAA 229 AAA 230 AAA 231 AAA 232 AAA 233 AAA 234 AAA 235 AAA 236 AAA 237 AAA 238 AAA 239 AAA 24 0 AAA 24 1 AAA 24 2 AAA 24 3 AAA 24 4 AAA 24 5 AAA 24 6 AAA 24 7 AAA 24 8 AAA 24 9 AAA 250 AAA 251 AAA 252 AAA 253 AAA 254 AAA 255 AAA 256 AAA 257 AAA 258 AAA 259 AAA 26 0 AAA 26 1 AAA 26 2 AAA 26 3 AAA 26 4 AAA 26 5 AAA 26 6 AAA 26 7 AAA 26 8 AAA 26 9 AAA 270 AAA 271 AAA 272 AAA 273 AAA 274 AAA 275 AAA 276 AAA 277 AAA 278 AAA 279 AAA 280 AAA 281 AAA 282 AAA 283 AAA 284 AAA 285 AAA 286 AAA 287 AAA 288 AAA 289 AAA 29 0 AAA 29 1 AAA 29 2 AAA 29 3 AAA 29 4 AAA 29 5 AAA 29 6 AAA 29 7 AAA 29 8 AAA 29 9 AAA 300 AAA 301 AAA 302 AAA 303 AAA 304 AAA 305 AAA 306 AAA 307 AAA 308 AAA 309 AAA 310 AAA 311 AAA 312 AAA 313 AAA 314 AAA 315 AAA 316 AAA 317 AAA 318 AAA 319 AAA 320 AAA 321 AAA 322 AAA 323 AAA 324 AAA 325 AAA 326 AAA 327 AAA 328 AAA 329 AAA 330 AAA 331 AAA 332 AAA 333 AAA 334 AAA 335 AAA 336 AAA 337 AAA 338 AAA 339 AAA 34 0 AAA 34 1 AAA 34 2 AAA 34 3 AAA 34 4 AAA 34 5 AAA 34 6 AAA 34 7 AAA 34 8 AAA 34 9 AAA 350 AAA 351 AAA 352 AAA 353 AAA 354 AAA 355 AAA 356 AAA 357 AAA 358 AAA 359 AAA 36 0 AAA 36 1 AAA 36 2 AAA 36 3 AAA 36 4 AAA 36 5 AAA 36 6 AAA 36 7 AAA 36 8 AAA 36 9 AAA 370 AAA 371 AAA 372 AAA 373 AAA 374 AAA 375 AAA 376 AAA 377 AAA 378 AAA 379 AAA 380 AAA 381 AAA 382 AAA 383 AAA 384 AAA 385 AAA 386 AAA 387 AAA 388 AAA 389 AAA 39 0 AAA 39 1 AAA 39 2 AAA 39 3 AAA 39 4 AAA 39 5 AAA 39 6 AAA 39 7 AAA 39 8 AAA 39 9 AAA 4 00 AAA 4 01 AAA 4 02 AAA 4 03 AAA 4 04 AAA 4 05 AAA 4 06 AAA 4 07 AAA 4 08 AAA 4 09 AAA 4 10 AAA 4 11 AAA 4 12 AAA 4 13 AAA 4 14 AAA 4 15 AAA 4 16 AAA 4 17 AAA 4 18 AAA 4 19 AAA 4 20 AAA 4 21 AAA 4 22 AAA 4 23 AAA 4 24 AAA 4 25 AAA 4 26 AAA 4 27 AAA 4 28 AAA 4 29 AAA 4 30 AAA 4 31 AAA 4 32 AAA 4 33 AAA 4 34 AAA 4 35 AAA 4 36 AAA 4 37 AAA 4 38 AAA 4 39 AAA 4 4 0 AAA 4 4 1 AAA 4 4 2 AAA 4 4 3 AAA 4 4 4 AAA 4 4 5 AAA 4 4 6 AAA 4 4 7 AAA 4 4 8 AAA 4 4 9 AAA 4 50 AAA 4 51 AAA 4 52 AAA 4 53 AAA 4 54 AAA 4 55 AAA 4 56 AAA 4 57 AAA 4 58 AAA 4 59 AAA 4 6 0 AAA 4 6 1 AAA 4 6 2 AAA 4 6 3 AAA 4 6 4 AAA 4 6 5 AAA 4 6 6 AAA 4 6 7 AAA 4 6 8 AAA 4 6 9 AAA 4 70 AAA 4 71 AAA 4 72 AAA 4 73 AAA 4 74 AAA 4 75 AAA 4 76 AAA 4 77 AAA 4 78 AAA 4 79 AAA 4 80 AAA 4 81 AAA 4 82 AAA 4 83 AAA 4 84 AAA 4 85 AAA 4 86 AAA 4 87 AAA 4 88 AAA 4 89 AAA 4 9 0 AAA 4 9 1 AAA 4 9 2 AAA 4 9 3 AAA 4 9 4 AAA 4 9 5 AAA 4 9 6 AAA 4 9 7 AAA 4 9 8 AAA 4 9 9 AAA 500 AAA 501 AAA 502 AAA 503 AAA 504 AAA 505 AAA 506 AAA 507 AAA 508 AAA 509 AAA 510 AAA 511 AAA 512 AAA 513 AAA 514 AAA 515 AAA 516 AAA 517 AAA 518 AAA 519 AAA 520 AAA 521 AAA 522 AAA 523 AAA 524 AAA 525 AAA 526 AAA 527 AAA 528 AAA 529 AAA 530 AAA 531 AAA 532 AAA 533 AAA 534 AAA 535 AAA 536 AAA 537 AAA 538 AAA 539 AAA 54 0 AAA 54 1 AAA 54 2 AAA 54 3 AAA 54 4 AAA 54 5 AAA 54 6 AAA 54 7 AAA 54 8 AAA 54 9 AAA 550 AAA 551 AAA 552 AAA 553 AAA 554 AAA 555 AAA 556 AAA 557 AAA 558 AAA 559 AAA 56 0 AAA 56 1 AAA 56 2 AAA 56 3 AAA 56 4 AAA 56 5 AAA 56 6 AAA 56 7 AAA 56 8 AAA 56 9 AAA 570 AAA 571 AAA 572 AAA 573 AAA 574 AAA 575 AAA 576 AAA 577 AAA 578 AAA 579 AAA 580 AAA 581 AAA 582 AAA 583 AAA 584 AAA 585 AAA 586 AAA 587 AAA 588 AAA 589 AAA 59 0 AAA 59 1 AAA 59 2 AAA 59 3 AAA 59 4 AAA 59 5 AAA 59 6 AAA 59 7 AAA 59 8 AAA 59 9 AAA 6 00 AAA 6 01 AAA 6 02 AAA 6 03 AAA 6 04 AAA 6 05 AAA 6 06 AAA 6 07 AAA 6 08 AAA 6 09 AAA 6 10 AAA 6 11 AAA 6 12 AAA 6 13 AAA 6 14 AAA 6 15 AAA 6 16 AAA 6 17 AAA 6 18 AAA 6 19 AAA 6 20 AAA 6 21 AAA 6 22 AAA 6 23 AAA 6 24 AAA 6 25 AAA 6 26 AAA 6 27 AAA 6 28 AAA 6 29 AAA 6 30 AAA 6 31 AAA 6 32 AAA 6 33 AAA 6 34 AAA 6 35 AAA 6 36 AAA 6 37 AAA 6 38 AAA 6 39 AAA 6 4 0 AAA 6 4 1 AAA 6 4 2 AAA 6 4 3 AAA 6 4 4 AAA 6 4 5 AAA 6 4 6 AAA 6 4 7 AAA 6 4 8 AAA 6 4 9 AAA 6 50 AAA 6 51 AAA 6 52 AAA 6 53 AAA 6 54 AAA 6 55 AAA 6 56 AAA 6 57 AAA 6 58 AAA 6 59 AAA 6 6 0 AAA 6 6 1 AAA 6 6 2 AAA 6 6 3 AAA 6 6 4 AAA 6 6 5 AAA 6 6 6 AAA 6 6 7 AAA 6 6 8 AAA 6 6 9 AAA 6 70 AAA 6 71 AAA 6 72 AAA 6 73 AAA 6 74 AAA 6 75 AAA 6 76 AAA 6 77 AAA 6 78 AAA 6 79 AAA 6 80 AAA 6 81 AAA 6 82 AAA 6 83 AAA 6 84 AAA 6 85 AAA 6 86 AAA 6 87 AAA 6 88 AAA 6 89 AAA 6 9 0 AAA 6 9 1 AAA 6 9 2 AAA 6 9 3 AAA 6 9 4 AAA 6 9 5 AAA 6 9 6 AAA 6 9 7 AAA 6 9 8 AAA 6 9 9 AAA 700 AAA 701 AAA 702 AAA 703 AAA 704 AAA 705 AAA 706 AAA 707 AAA 708 AAA 709 AAA 710 AAA 711 AAA 712 AAA 713 AAA 714 AAA 715 AAA 716 AAA 717 AAA 718 AAA 719 AAA 720 AAA 721 AAA 722 AAA 723 AAA 724 AAA 725 AAA 726 AAA 727 AAA 728 AAA 729 AAA 730 AAA 731 AAA 732 AAA 733 AAA 734 AAA 735 AAA 736 AAA 737 AAA 738 AAA 739 AAA 74 0 AAA 74 1 AAA 74 2 AAA 74 3 AAA 74 4 AAA 74 5 AAA 74 6 AAA 74 7 AAA 74 8 AAA 74 9 AAA 750 AAA 751 AAA 752 AAA 753 AAA 754 AAA 755 AAA 756 AAA 757 AAA 758 AAA 759 AAA 76 0 AAA 76 1 AAA 76 2 AAA 76 3 AAA 76 4 AAA 76 5 AAA 76 6 AAA 76 7 AAA 76 8 AAA 76 9 AAA 770 AAA 771 AAA 772 AAA 773 AAA 774 AAA 775 AAA 776 AAA 777 AAA 778 AAA 779 AAA 780 AAA 781 AAA 782 AAA 783 AAA 784 AAA 785 AAA 786 AAA 787 AAA 788 AAA 789 AAA 79 0 AAA 79 1 AAA 79 2 AAA 79 3 AAA 79 4 AAA 79 5 AAA 79 6 AAA 79 7 AAA 79 8 AAA 79 9 AAA 800 AAA 801 AAA 802 AAA 803 AAA 804 AAA 805 AAA 806 AAA 807 AAA 808 AAA 809 AAA 810 AAA 811 AAA 812 AAA 813 AAA 814 AAA 815 AAA 816 AAA 817 AAA 818 AAA 819 AAA 820 AAA 821 AAA 822 AAA 823 AAA 824 AAA 825 AAA 826 AAA 827 AAA 828 AAA 829 AAA 830 AAA 831 AAA 832 AAA 833 AAA 834 AAA 835 AAA 836 AAA 837 AAA 838 AAA 839 AAA 84 0 AAA 84 1 AAA 84 2 AAA 84 3 AAA 84 4 AAA 84 5 AAA 84 6 AAA 84 7 AAA 84 8 AAA 84 9 AAA 850 AAA 851 AAA 852 AAA 853 AAA 854 AAA 855 AAA 856 AAA 857 AAA 858 AAA 859 AAA 86 0 AAA 86 1 AAA 86 2 AAA 86 3 AAA 86 4 AAA 86 5 AAA 86 6 AAA 86 7 AAA 86 8 AAA 86 9 AAA 870 AAA 871 AAA 872 AAA 873 AAA 874 AAA 875 AAA 876 AAA 877 AAA 878 AAA 879 AAA 880 AAA 881 AAA 882 AAA 883 AAA 884 AAA 885 AAA 886 AAA 887 AAA 888 AAA 889 AAA 89 0 AAA 89 1 AAA 89 2 AAA 89 3 AAA 89 4 AAA 89 5 AAA 89 6 AAA 89 7 AAA 89 8 AAA 89 9 AAA 9 00 AAA 9 01 AAA 9 02 AAA 9 03 AAA 9 04 AAA 9 05 AAA 9 06 AAA 9 07 AAA 9 08 AAA 9 09 AAA 9 10 AAA 9 11 AAA 9 12 AAA 9 13 AAA 9 14 AAA 9 15 AAA 9 16 AAA 9 17 AAA 9 18 AAA 9 19 AAA 9 20 AAA 9 21 AAA 9 22 AAA 9 23 AAA 9 24 AAA 9 25 AAA 9 26 AAA 9 27 AAA 9 28 AAA 9 29 AAA 9 30 AAA 9 31 AAA 9 32 AAA 9 33 AAA 9 34 AAA 9 35 AAA 9 36 AAA 9 37 AAA 9 38 AAA 9 39 AAA 9 4 0 AAA 9 4 1 AAA 9 4 2 AAA 9 4 3 AAA 9 4 4 AAA 9 4 5 AAA 9 4 6 AAA 9 4 7 AAA 9 4 8 AAA 9 4 9 AAA 9 50 AAA 9 51 AAA 9 52 AAA 9 53 AAA 9 54 AAA 9 55 AAA 9 56 AAA 9 57 AAA 9 58 AAA 9 59 AAA 9 6 0 AAA 9 6 1 AAA 9 6 2 AAA 9 6 3 AAA 9 6 4 AAA 9 6 5 AAA 9 6 6 AAA 9 6 7 AAA 9 6 8 AAA 9 6 9 AAA 9 70 AAA 9 71 AAA 9 72 AAA 9 73 AAA 9 74 AAA 9 75 AAA 9 76 AAA 9 77 AAA 9 78 AAA 9 79 AAA 9 80 AAA 9 81 AAA 9 82 AAA 9 83 AAA 9 84 AAA 9 85 AAA 9 86 AAA 9 87 AAA 9 88 AAA 9 89 AAA 9 9 0 AAA 9 9 1 AAA 9 9 2 AAA 9 9 3 AAA 9 9 4 AAA 9 9 5 AAA 9 9 6 AAA 9 9 7 AAA 9 9 8 AAA 9 9 9 AAA 000 AAB 001 AAB 002 AAB 003 AAB 004 AAB 005 AAB 0A 26 PO SSIBLE CHARACTERS HERE = = > 0 Sta rt of wha t full list would look like 26 PO SSIBLE CHARACTERS HERE = = > Use the mul ti pl icatio n Rul e beca use du pli cates are o k. Where as with Co mbin atio ns and Pe rmutation s, you ca n't re pea t the sa me chara cte r m ore th an on ce (unl ess it is orig ina ll y l isted more th an once , li ke Blu e, Bl ue, R ed). If you used the two permuta tion formula s, multiplying them together would not give you a ll the possibilities Permuta tions of Numbers 720 Permuta tions of Letters 15,6 00 product 11,232,000 A A 289 AAC 29 0 AAC 29 1 AAC 29 2 AAC 29 3 AAC 29 4 AAC 29 5 AAC 29 6 AAC 29 7 AAC 29 8 AAC 29 9 AAC 300 AAC 301 AAC 302 AAC 303 AAC 304 AAC 305 AAC 306 AAC 307 AAC 308 AAC 309 AAC 310 AAC 311 AAC 312 AAC 313 AAC 314 AAC 315 AAC 316 AAC 317 AAC 318 AAC 319 AAC 320 AAC 321 AAC 322 AAC 323 AAC 324 AAC 325 AAC 326 AAC 327 AAC 328 AAC 329 AAC 330 AAC 331 AAC 332 AAC 333 AAC 334 AAC 335 AAC 336 AAC 337 AAC 338 AAC 339 AAC 34 0 AAC 34 1 AAC 34 2 AAC 34 3 AAC 34 4 AAC 34 5 AAC 34 6 AAC 34 7 AAC 34 8 AAC 34 9 AAC 350 AAC 351 AAC 352 AAC 353 AAC 354 AAC 355 AAC 356 AAC 357 AAC 358 AAC 359 AAC 36 0 AAC 36 1 AAC 36 2 AAC 36 3 AAC 36 4 AAC 36 5 AAC 36 6 AAC 36 7 AAC 36 8 AAC 36 9 AAC 370 AAC 371 AAC 372 AAC 373 AAC 374 AAC 375 AAC 376 AAC 377 AAC 378 AAC 379 AAC 380 AAC 381 AAC 382 AAC 383 AAC 384 AAC 385 AAC 386 AAC 387 AAC 388 AAC 389 AAC 39 0 AAC 39 1 AAC 39 2 AAC 39 3 AAC 39 4 AAC 39 5 AAC 39 6 AAC 39 7 AAC 39 8 AAC 39 9 AAC 4 00 AAC 4 01 AAC 4 02 AAC 4 03 AAC 4 04 AAC 4 05 AAC 4 06 AAC 4 07 AAC 4 08 AAC 4 09 AAC 4 10 AAC 4 11 AAC 4 12 AAC 4 13 AAC 4 14 AAC 4 15 AAC 4 16 AAC 4 17 AAC 4 18 AAC 4 19 AAC 4 20 AAC 4 21 AAC 4 22 AAC 4 23 AAC 4 24 AAC 4 25 AAC 4 26 AAC 4 27 AAC 4 28 AAC 4 29 AAC 4 30 AAC 4 31 AAC 4 32 AAC 4 33 AAC 4 34 AAC 4 35 AAC 4 36 AAC 4 37 AAC 4 38 AAC 4 39 AAC 4 4 0 AAC 4 4 1 AAC 4 4 2 AAC 4 4 3 AAC 4 4 4 AAC 4 4 5 AAC 4 4 6 AAC 4 4 7 AAC 4 4 8 AAC 4 4 9 AAC 4 50 AAC 4 51 AAC 4 52 AAC 4 53 AAC 4 54 AAC 4 55 AAC 4 56 AAC 4 57 AAC 4 58 AAC 4 59 AAC 4 6 0 AAC 4 6 1 AAC 4 6 2 AAC 4 6 3 AAC 4 6 4 AAC 4 6 5 AAC 4 6 6 AAC 4 6 7 AAC 4 6 8 AAC 4 6 9 AAC 4 70 AAC 4 71 AAC 4 72 AAC 4 73 AAC 4 74 AAC 4 75 AAC 4 76 AAC 4 77 AAC 4 78 AAC 4 79 AAC 4 80 AAC 4 81 AAC 4 82 AAC 4 83 AAC 4 84 AAC 4 85 AAC 4 86 AAC 4 87 AAC 4 88 AAC 4 89 AAC 4 9 0 AAC 4 9 1 AAC 4 9 2 AAC 4 9 3 AAC 4 9 4 AAC 4 9 5 AAC 4 9 6 AAC 4 9 7 AAC 4 9 8 AAC 4 9 9 AAC 500 AAC 501 AAC 502 AAC 503 AAC 504 AAC 505 AAC 506 AAC 507 AAC 508 AAC 509 AAC 510 AAC 511 AAC 512 AAC 513 AAC 514 AAC 515 AAC 516 AAC 517 AAC 518 AAC 519 AAC 520 AAC 521 AAC 522 AAC 523 AAC 524 AAC 525 AAC 526 AAC 527 AAC 528 AAC 529 AAC 530 AAC 531 AAC 532 AAC 533 AAC 534 AAC 535 AAC 536 AAC 537 AAC 538 AAC 539 AAC 54 0 AAC 54 1 AAC 54 2 AAC 54 3 AAC 54 4 AAC 54 5 AAC 54 6 AAC 54 7 AAC 54 8 AAC 54 9 AAC 550 AAC 551 AAC 552 AAC 553 AAC 554 AAC 555 AAC 556 AAC 557 AAC 558 AAC 559 AAC 56 0 AAC 56 1 AAC 56 2 AAC 56 3 AAC 56 4 AAC 56 5 AAC 56 6 AAC 56 7 AAC 56 8 AAC 56 9 AAC 570 AAC 571 AAC 572 AAC 573 AAC 574 AAC 575 AAC 576 AAC 577 AAC 578 AAC 579 AAC 580 AAC 581 AAC 582 AAC 583 AAC 584 AAC 585 AAC 586 AAC 587 AAC 588 AAC 589 AAC 59 0 AAC 59 1 AAC 59 2 AAC 59 3 AAC 59 4 AAC 59 5 AAC 59 6 AAC 59 7 AAC 59 8 AAC 59 9 AAC 6 00 AAC 6 01 AAC 6 02 AAC 6 03 AAC 6 04 AAC 6 05 AAC 6 06 AAC 6 07 AAC 6 08 AAC 6 09 AAC 6 10 AAC 6 11 AAC 6 12 AAC 6 13 AAC 6 14 AAC 6 15 AAC 6 16 AAC 6 17 AAC 6 18 AAC 6 19 AAC 6 20 AAC 6 21 AAC 6 22 AAC 6 23 AAC 6 24 AAC 6 25 AAC 6 26 AAC 6 27 AAC 6 28 AAC 6 29 AAC 6 30 AAC 6 31 AAC 6 32 AAC 6 33 AAC 6 34 AAC 6 35 AAC 6 36 AAC 6 37 AAC 6 38 AAC 6 39 AAC 6 4 0 AAC 6 4 1 AAC 6 4 2 AAC 6 4 3 AAC 6 4 4 AAC 6 4 5 AAC 6 4 6 AAC 6 4 7 AAC 6 4 8 AAC 6 4 9 AAC 6 50 AAC 6 51 AAC 6 52 AAC 6 53 AAC 6 54 AAC 6 55 AAC 6 56 AAC 6 57 AAC 6 58 AAC 6 59 AAC 6 6 0 AAC 6 6 1 AAC 6 6 2 AAC 6 6 3 AAC 6 6 4 AAC 6 6 5 AAC 6 6 6 AAC 6 6 7 AAC 6 6 8 AAC 6 6 9 AAC 6 70 AAC 6 71 AAC 6 72 AAC 6 73 AAC 6 74 AAC 6 75 AAC 6 76 AAC 6 77 AAC 6 78 AAC 6 79 AAC 6 80 AAC 6 81 AAC 6 82 AAC 6 83 AAC 6 84 AAC 6 85 AAC 6 86 AAC 6 87 AAC 6 88 AAC 6 89 AAC 6 9 0 AAC 6 9 1 AAC 6 9 2 AAC 6 9 3 AAC 6 9 4 AAC 6 9 5 AAC 6 9 6 AAC 6 9 7 AAC 6 9 8 AAC 6 9 9 AAC 700 AAC 701 AAC 702 AAC 703 AAC 704 AAC 705 AAC 706 AAC 707 AAC 708 AAC 709 AAC 710 AAC 711 AAC 712 AAC 713 AAC 714 AAC 715 AAC 716 AAC 717 AAC 718 AAC 719 AAC 720 AAC 721 AAC 722 AAC 723 AAC 724 AAC 725 AAC 726 AAC 727 AAC 728 AAC 729 AAC 730 AAC 731 AAC 732 AAC 733 AAC 734 AAC 735 AAC 736 AAC 737 AAC 738 AAC 739 AAC 74 0 AAC 74 1 AAC 74 2 AAC 74 3 AAC 74 4 AAC 74 5 AAC 74 6 AAC 74 7 AAC 74 8 AAC 74 9 AAC 750 AAC 751 AAC 752 AAC 753 AAC 754 AAC 755 AAC 756 AAC 757 AAC 758 AAC 759 AAC 76 0 AAC 76 1 AAC 76 2 AAC 76 3 AAC 76 4 AAC 76 5 AAC 76 6 AAC 76 7 AAC 76 8 AAC 76 9 AAC 770 AAC 771 AAC 772 AAC 773 AAC 774 AAC 775 AAC 776 AAC 777 AAC 778 AAC 779 AAC 780 AAC 781 AAC 782 AAC 783 AAC 784 AAC 785 AAC 786 AAC 787 AAC 788 AAC 789 AAC 79 0 AAC 79 1 AAC 79 2 AAC 79 3 AAC 79 4 AAC 79 5 AAC 79 6 AAC 79 7 AAC 79 8 AAC 79 9 AAC 800 AAC 801 AAC 802 AAC 803 AAC 804 AAC 805 AAC 806 AAC 807 AAC 808 AAC 809 AAC 810 AAC 811 AAC 812 AAC 813 AAC 814 AAC 815 AAC 816 AAC 817 AAC 818 AAC 819 AAC 820 AAC 821 AAC 822 AAC 823 AAC 824 AAC 825 AAC 826 AAC 827 AAC 828 AAC 829 AAC 830 AAC 831 AAC 832 AAC 833 AAC 834 AAC 835 AAC 836 AAC 837 AAC 838 AAC 839 AAC 84 0 AAC 84 1 AAC 84 2 AAC 84 3 AAC 84 4 AAC 84 5 AAC 84 6 AAC 84 7 AAC 84 8 AAC 84 9 AAC 850 AAC 851 AAC 852 AAC 853 AAC 854 AAC 855 AAC 856 AAC 857 AAC 858 AAC 859 AAC 86 0 AAC 86 1 AAC 86 2 AAC 86 3 AAC 86 4 AAC 86 5 AAC 86 6 AAC 86 7 AAC 86 8 AAC 86 9 AAC 870 AAC 871 AAC 872 AAC 873 AAC 874 AAC 875 AAC 876 AAC 877 AAC 878 AAC 879 AAC 880 AAC 881 AAC 882 AAC 883 AAC 884 AAC 885 AAC 886 AAC 887 AAC 888 AAC 889 AAC 89 0 AAC 89 1 AAC 89 2 AAC 89 3 AAC 89 4 AAC 89 5 AAC 89 6 AAC 89 7 AAC 89 8 AAC 89 9 AAC 9 00 AAC 9 01 AAC 9 02 AAC 9 03 AAC 9 04 AAC 9 05 AAC 9 06 AAC 9 07 AAC 9 08 AAC 9 09 AAC 9 10 AAC 9 11 AAC 9 12 AAC 9 13 AAC 9 14 AAC 9 15 AAC 9 16 AAC 9 17 AAC 9 18 AAC 9 19 AAC 9 20 AAC 9 21 AAC 9 22 AAC 9 23 AAC 9 24 AAC 9 25 AAC 9 26 AAC 9 27 AAC 9 28 AAC 9 29 AAC 9 30 AAC 9 31 AAC 9 32 AAC 9 33 AAC 9 34 AAC 9 35 AAC 9 36 AAC 9 37 AAC 9 38 AAC 9 39 AAC 9 4 0 AAC 9 4 1 AAC 9 4 2 AAC 9 4 3 AAC 9 4 4 AAC 9 4 5 AAC 9 4 6 AAC 9 4 7 AAC 9 4 8 AAC 9 4 9 AAC 9 50 AAC 9 51 AAC 9 52 AAC 9 53 AAC 9 54 AAC 9 55 AAC 9 56 AAC 9 57 AAC 9 58 AAC 9 59 AAC 9 6 0 AAC 9 6 1 AAC 9 6 2 AAC 9 6 3 AAC 9 6 4 AAC 9 6 5 AAC 9 6 6 AAC 9 6 7 AAC 9 6 8 AAC 9 6 9 AAC 9 70 AAC 9 71 AAC 9 72 AAC 9 73 AAC 9 74 AAC 9 75 AAC 9 76 AAC 9 77 AAC 9 78 AAC 9 79 AAC 9 80 AAC 9 81 AAC 9 82 AAC 9 83 AAC 9 84 AAC 9 85 AAC 9 86 AAC 9 87 AAC 9 88 AAC 9 89 AAC 9 9 0 AAC 9 9 1 AAC 9 9 2 AAC 9 9 3 AAC 9 9 4 AAC 9 9 5 AAC 9 9 6 AAC 9 9 7 AAC 9 9 8 AAC 9 9 9 AAC 1000 AAD 1001 AAC 1002 AAC 1003 AAC 1004 AAC 1005 AAC 1006 AAC 1007 AAC 1008 AAC 1009 AAC 1010 AAC 1011 AAC 1012 AAC 1013 AAC 1014 AAC 1015 AAC 1016 AAC 1017 AAC 1018 AAC 1019 AAC 1020 AAC 1021 AAC 1022 AAC 1023 AAC 1024 AAC 1025 AAC 1026 AAC 1027 AAC 1028 AAC 1029 AAC 1030 AAC 1031 AAC 1032 AAC 1033 AAC 1034 AAC 1035 AAC 1036 AAC 1037 AAC 1038 AAC 1039 AAC 104 0 AAC 104 1 AAC 104 2 AAC 104 3 AAC 104 4 AAC 104 5 AAC 104 6 AAC 104 7 AAC 104 8 AAC 104 9 AAC 1050 AAC 1051 AAC 1052 AAC 1053 AAC 1054 AAC 1055 AAC 1056 AAC 1057 AAC 1058 AAC 1059 AAC 106 0 AAC 106 1 AAC 106 2 AAC 106 3 AAC 106 4 AAC 106 5 AAC 106 6 AAC 106 7 AAC 106 8 AAC 106 9 AAC 1070 AAC 1071 AAC 1072 AAC 1073 AAC 1074 AAC 1075 AAC 1076 AAC 1077 AAC 1078 AAC 1079 AAC 1080 AAC 1081 AAC 1082 AAC 1083 AAC 1084 AAC 1085 AAC 1086 AAC 1087 AAC 1088 AAC 1089 AAC 109 0 AAC 109 1 AAC 109 2 AAC 109 3 AAC 109 4 AAC 109 5 AAC 109 6 AAC 109 7 AAC 109 8 AAC 109 9 AAC 1100 AAC 1101 AAC 1102 AAC 1103 AAC 1104 AAC 1105 AAC 1106 AAC 1107 AAC 1108 AAC 1109 AAC 1110 AAC 1111 AAC 1112 AAC 1113 AAC 1114 AAC 1115 AAC 1116 AAC 1117 AAC 1118 AAC 1119 AAC 1120 AAC 1121 AAC 1122 AAC 1123 AAC 1124 AAC 1125 AAC 1126 AAC 1127 AAC 1128 AAC 1129 AAC 1130 AAC 1131 AAC 1132 AAC 1133 AAC 1134 AAC 1135 AAC 1136 AAC 1137 AAC 1138 AAC 1139 AAC 114 0 AAC 114 1 AAC 114 2 AAC 114 3 AAC 114 4 AAC 114 5 AAC 114 6 AAC 114 7 AAC 114 8 AAC 114 9 AAC 1150 AAC 1151 AAC 1152 AAC 1153 AAC 1154 AAC 1155 AAC 1156 AAC 1157 AAC 1158 AAC 1159 AAC 116 0 AAC 116 1 AAC 116 2 AAC 116 3 AAC 116 4 AAC 116 5 AAC 116 6 AAC 116 7 AAC 116 8 AAC 116 9 AAC 1170 AAC 1171 AAC 1172 AAC 1173 AAC 1174 AAC 1175 AAC 1176 AAC 1177 AAC 1178 AAC 1179 AAC 1180 AAC 1181 AAC 1182 AAC 1183 AAC 1184 AAC 1185 AAC 1186 AAC 1187 AAC 1188 AAC 1189 AAC 119 0 AAC Total > 60 years of age Female Female and > 60 years of age Male Male and > 60 years of age < 60 years of age Total # of people being considered = > 60 years of age # of people being considered = Female # of people being considered = Female and > 60 years of age # of people being considered = Male # of people being considered = Male and > 60 years of age # of people being considered = < 60 years of age # of people being considered = 4 3 2 1 2 2 1 P(> 60 years of age) P(Female) P(Female and > 60 years of age) P(Male) P(Male and > 60 years of age) P(< 60 years of age) < 60 years of age | Male Female | > 60 years of age < 60 years of age | Male # of people being considered = Female | > 60 years of age # of people being considered = 0 1 P(< 60 years of age | Male) P(Female | > 60 years of age) 0.75 0.5 0.25 0.5 0.5 0.25 0 0.333 0.25 H N a # of Holly Land tracts purchased = # of Newburg Land tracts purchased = 2 P(next 2 tracts sold are N) = 0 4 Trials P(next 4 tracts sold 0 are H) = b P(next 4 tracts at least 1 is H) = P(next 4 tracts at least 1 is H) = c Dependent Success = Sell H Sell H Not Sell H # of Outcomes = # of Trials Sample Space = Outcomes ^ (# of Trials) 4 P(H) 6 P(N) 10 0.4 0.6 0.3333 = 6/10*5/9 0.0714285714 0.92857143 0.92857143 1 SH 1 NSH 2 4 16 Try to sell Tract (S = SH) Count of Successes (S = SH) Probability of Occurrence 1 2 3 4 1 SH SH SH SH 4 0.00476190 2 NSH SH SH SH 3 0.02857143 3 SH NSH SH SH 3 0.02857143 4 SH SH NSH SH 3 0.02857143 5 SH SH SH NSH 3 0.02857143 6 NSH NSH SH SH 2 0.07142857 7 SH NSH NSH SH 2 0.07142857 8 SH SH NSH NSH 2 0.07142857 9 NSH SH NSH SH 2 0.07142857 10 NSH SH SH NSH 2 0.07142857 11 SH NSH SH NSH 2 0.07142857 12 SH NSH NSH NSH 1 0.09523810 13 NSH SH NSH NSH 1 0.09523810 14 NSH NSH SH NSH 1 0.09523810 15 NSH NSH NSH SH 1 0.09523810 16 NSH NSH NSH NSH 0 0.07142857 1.00000000 Possible # Outcomes 6 POSSIBLE CHARACTERS HERE ==> 456,976 6 POSSIBLE CHARACTERS HERE ==> 4 26 6 POSSIBLE CHARACTERS HERE ==> Characters available for password = Letters available for each of the 4 characters = Each Character can be used more than once # of Passwords Available 6 POSSIBLE CHARACTERS HERE ==> # # # # Total Items = Size of Sub Group = Conditional Probability = X/Y*(X-1)/(Y-1) n r 5 3 Permutations = 60 5*4*3*2*1 3*3*1 5-3 = 2 …… 2*1 120 6 2 120/2 = (5*4*3*2*1)/(5-3 = 2 …… 2*1) 3/5*2/4*1/3 120 6 60 0.1000 0.1000 Total cans in case Cans that are contaminated 1 Trial = Pull 1 can # of Trials = 24 n 1 3r # of combinations of cans that could be selected 2,024 P(Can Not Selected during 3 trials) P(contaminated can is selected during Experiment) 0.875 0.125 SC, NSC, NSC NSC, SC, NSC NSC, NSC, SC P(contaminated can is selected during Experiment) 0.0416667 0.0416667 0.0416667 0.125 ...
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This note was uploaded on 01/10/2012 for the course BUSN 210 taught by Professor Girvin during the Winter '09 term at Des Moines CC.

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