Week4-2up - Mutually exclusive (disjoint) p. 118 Additive...

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Unformatted text preview: Mutually exclusive (disjoint) p. 118 Additive Rule of Probability p. 117 Complement p. 115 : ¢¤ £ Conditional probability p. 122  ¥ ¦ ¢¤ £ ¡ ¢¤ £ ¥ ¡ LAST TIME : be asked about the INTERPRETATION of the graphs. similar graphs (different data set)–similar questions will PROJECT 1 : NOT COLLECTED!!! BUT–on next quiz, For Wednesday : p. 127–135, Read pages 163 – 170 3.55–3.57, 3.59, 3.60, 3.61 For tomorrow: Probs : 3.47, 3.49, 3.50, 3.52, 3.53, For Today : p. 127–135 Assignments you’re not in shape for it, it’s too far to walk back. The trouble with jogging is that, by the time you realize STA 2023 c B.Presnell & D.Wackerly - Lecture 6 ¥  ¢£ ¡ ¡ ¥ ¨ ¢¤ £ ¢¤ £  ¥ ¦ ¢ ¢ £ ¤£ ¥   ¥ 72 EXAMPLE : Pol. party – job type. STA 2023 c B.Presnell & D.Wackerly - Lecture 6 ¡¤ ¡ ¡ person is a white collar worker .  RB RW DB DW IB IW probability that (s)he is a white collar worker? If I know that a person is a Republican, what is the person is a Republican . ¢¤ £  ¥ ¦ § ¨ ¢¤ £  © ¥  ¤ ¢£ ¢£ ¥ !¥ ¦ "'  ¦ §" probability that (s)he is a blue collar worker? If a person is NOT a Democrat, what is the ¦ ¥ ¢£ ¥  ¢¤ £ ¥  #¢ £ #¢ £ !¥ ¦ ¦ ) ¥  ¡  ¦  ¦   §"  '¦ $¢ £ $¢ £ §" '( !¥ ¥ ¦' ' ( 73 %& ¦ ¦ %( Simple Events ¤ ¦ ¤ Given: individual smokes develops cancer Probabilities  "¦ )% ¢£ ¤  ¥ ¦ ¢ ¢¤ £  £ ¥ ¤ ¥ given that the person smokes. Find the probability that a person develops cancer % EXAMPLE 3.13: (p. 123) ¤  ©  © ¤  ¤   ©  © "   ) 74 ¦"  ¦ )% %  ¦" % Multiplicative Law of Prob. (p. 128) STA 2023 c B.Presnell & D.Wackerly - Lecture 6 so Why? Recall (Defn of Cond‘l Prob.): ¢£ 75 Conditional Probability is given when : Useful for computing the probability of the intersection ¤  %' ¦ ¢£ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 ) ¥ ¢¤ £ ¢¤ £ ¥ " ¢¤ £ ¢ ¤£ ©¥ ¥ ¦ ¦ %' ¢¤ £   " ¥ ¦  ) ¢¤ £ ¦ § ¨  ¢¤ £ ©¥ ¥ ¦ ¢¤ £ ¤  ¥ ¥ ¦ ¢¤ £ ¦ )% ¢¤ £  ¥ ¦ ¢¤ £ ¢¤ £ ¢¤ £ ¢¤ £ Conditional Probability easy to get without definition. ¡  " ¥ ¥ ¥ ¡ ¥ ¢£ ¥ ¢£ ¦ ¤  ¥ ¥ ¥ ¢¤ £ ¥ ¤¢ £  ¥ ¡ ¦ ¢¤ £  ¥ ¢£  ¦ ¦ ¥ Chip lasts from first use through first year , .  ¥ “Does not fail during first year” ¢£ ¦ ¢¤ £ ¡  ¤  ¡ ¡ ¢£  ¤  ¦  % Microchip lasts through first use , ¡¤ ¢¤ £ ¥ ¦ on its first use!!! NOTE: If chip does not fail during first year–It cannot fail during the first year. . Find the probability that a microchip does not fail ¥ ¦ . 77 : : there are 29 cameras, 23 good, 6 bad. when we select the second, Note that shipment accepted . Start with 30 cameras, 6 bad. If first chosen is good, camera good second is good? If first camera known to be good, what is (cond.) prob. camera good good, accept shipment. Find the prob. accept shipment. cameras, 6 defective. Choose 2 at random. If both EXAMPLE : (Screening for quality) Shipment of 30 STA 2023 c B.Presnell & D.Wackerly - Lecture 6 ¤ ¡ ¡ is  ¢£ If it lasts through its first use, prob. that it lasts a year ¡ ¤ ¦" ¤  ¤ fail on first use ¢¤ £ ¡ §  EXAMPLE : (#3.61, p. 137) Microchip: ¥ ¦ ¥ ¦   ¦ ¥ ¥ ¦¤ % © ©   ¦ 76 %  ¦ ¢¤ £ ¢£ ¢¤ £  ¥ ¦ ¢¤ £ ¥ ¢£ ¤  ¥ '¦ " ( "' ¦  %§  ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 ¡  ¢ ¤   " £ § ¥ ¦" ¥ ¦ § ¨  ¦ &'  '( % ¦ ( Is ¤¢ £  ¥ ¦ ¡ ¡¤ ? and get an even number ¡ ¡ ¡ ¡ ¥ # greater than 3 # greater than 2 ¥ ¦ ¡  ¡¤  ¦ ¦    ¢¤ £ ¡¤  ¦ ¤ ¦ ¦ ¥ ¡ &(  ¡ , £  ¡  ¡ ¦  ©  &(  ¢£ ¦ § '"  &) ( EXAMPLE : Toss a balanced die. ¡ ¢¤ £ an impact on the probability of another event? Does knowing that one event has occured alway have Independence STA 2023 c B.Presnell & D.Wackerly - Lecture 6  and 78 STA 2023 c B.Presnell & D.Wackerly - Lecture 6 said to be independent if Note that if and and and Ex. Die Toss: “Not independent” = “DEPENDENT.” or DEFINITION: (Def. 3.9, p. 131) Events ¤  ¥ ¡   ¦ ¤ ¡ ¡   ¡ ¥¥ ¥ § ¡ ¢ ¢£ ' ¢¤ £ ¡ ¦ ¦ ¦ ¡ ¥  ¢¤ £ § ¥ ¡ ¦ ( &)  ¡ ¢£  £ ¦ ¤ ¦ " &(  ¤ ¤ ¤ ¢¤ £ ( &)  © ¤¢ £  ¥ ¤  ¥ ¦ ¢¤ £ ¦ INDEPENDENT. DEPENDENT,  ¥ and are [by independence] [by Mult. Law] are independent then ¢¤ £ ¦ ¤  ¢¤ £  ¥ ¥ ¢¤ £ ¦ © ¥ ¢£ ¢£ ¥ ¢£ ¤  ¥ ¥ ¢£ ¥ ¦ ¤ ¥ ¥ ¦ ¢¤ £ ¥ ¤¢ £  ¥ ¦ ¢¤ £ ¢£ ¥  ¥ ¦ ¦ ¥ ¥ © ¤ ¦ ¤ ¤ ¥ ¥ © ¤  ¤ '¦ " " ¦ §  ¦ ¢£   ¥ 79 sets have no points in Independent – need to check PROBABILITIES. common.  , are mutually exclusive ¢¤ £ ¤ and  Suppose that ¥ ¡ Mutually Exclusive ¢¤ £ ¦ ¢¤ £  ¥ and and that are DEPENDENT!!  ¦ ¡ EXCLUSIVE. Do NOT confuse INDEPENDENT with MUTUALLY WARNING Independent This is another way to define independence: (p. 133) § ¥ ¢¤ £ ¢£ ¥  ¢ % ¡ ¡ ¦ § ¢£ ¥ § ¦ ¢ 80 TTT TTH THT THH HTT HTH HHT HHH Outcome Probability assuming matches independent : in the example for 8 possible outcomes were calculated . T wins with prob. 81 . Probs. given Odds are 2:1 that H wins means H wins with prob. and winner of each recorded. play racquetball, H will win. H and T play 3 matches, Ex. (LAST TIME!!!) Odds are 2:1 that when H and T STA 2023 c B.Presnell & D.Wackerly - Lecture 6 ¥ ¢¤ £  ¥ ¦ " "£ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 ¢£  ¥  §¥ ¥ ¦ ¦ ¢¤ £ ¥ ¢¤ £ % ¦ ¢£ ¥ % ¥ ¥ ¥ ¥ ¥ ¥ ¥  ¥ '" £ '" £ '" £ '" £ §' £ §' £ §' £ ¦ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¤ ¥ ¥ '" £ '" £ §' £ §' £ '" £ '" £ ¥ §' £ ¥ ¥ ¥ §' £ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¡ ¡ ¡ ¡ ¡ ¥ ¥ ¥ ¥ ¥ ¡ ¡ '" £ §' £ '" £ §' £ '" £ §' £ ¡ §' £ ¡ '" £ §' £ ¥ ¥ ¥ ¦ % '" ¦ §' ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¦ ¥ ¥ ¥ ¥ ¥ ¥ ¦ ¦ ¦ ¦" ¥ ¢ " ¥ "( ¥ "( ¥ " ¥ "( ¦" ¦" " ¥ " ¥ "§ § ¤ Find ¥ ¢¤ £ ¦ ¤¢ £  ¥  ¥ ¦ ¢¤ £  ¢¤ £ independent events: ¥ ¢¤ £ ' "¦ § ¦ ¢¤ £ ¥ § ¥ ¢¤ £ ¦  ¡ ¥ ¢¤ £ ¥ "¦ § and ¢¤ £  ¥ ¢¤ £  ¥ ¡ ¢£ &¦ § ) (Add. Law) (independence) . (independence) ¥ ¡ ¡ ¢£ ¥ ¢£  ¥ ¨ ¦ ¥  ¢¤ £ ¢£  '¦ § ¢¤ £ Multiplicative Rule (p. 128): Last Time : QUIZ 2 – Chapter 3 – Project 1 For Tuesday:Exercises 4.22, 4.27-29, For Monday: Read pages 172–176, 179-183 3.114, 4.3–4.5, 4.7, 4.11, 4.14, 4.16 For tomorrow: Exercises 3.99, 3.100, 3.105-109, Today : p. 163–169 Assignments : can’t, you’re right! Thought: Whether you think you can or you think you ¡ ¡¤ ¥ '¦ " "¦ § '  § &¨ § &¦ ( and ¢£  ¢¤ £  Independent (p. 131, 133): ¤   ¥ ¦ ¥ ¦ ¢¤ £ ¢£ ¢¤ £ are ¡ and ¤  EXAMPLE : Like #3.52(a), p. 136 ¢£  ¥ ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 ¥ ¢¤ £  ¥ ¢¤ £ ¥ ¢£ ¡ ¥ ¦  ¢£ ¦ ¢¤ £  ¥ ¥ ¥ ¦ 82 ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 ¡ 83 ¢¤ £ ¥ ! ! . . . sixth field joint works ¥ at least 1 fails ¦ ¢ ¢ ¢ all six work Note: At ¡ , about ¢ -th of what it was at at least 1 fails £ , so F), estimate ¢ F. , only & ¢£ ¦ ¢£ Under Challenger flight conditions ( £! ¥ ¦ ¢£ all six work ¢£ ¢£ ! ! Assuming joint failures indep.: ¤ first field joint works  £ ¦  all six field joints (sealed by O-rings) work. ! £ ¥ © ¥  !  © ¢£ '  !¥ ¦ ¡ § ¢£ ¦ ¢£ £ !¥ § ¦ £¢ ¦ ¥¢ £ ¦ £ ¤ ! ¥ ¥ ¥¤  ¦ § ¦ ¢£  !  ¢ £ & % ¥ ¢£ ¦ £ !¥ § ' ¨ £ ¦ §& ¦ ¥¤ ¢ £  ¦ ¤ !¥ §%  ¦ ¨ ¢  ¦ !¥ ¡ ¢ ¢ § ¦' ¦   ! ¢£ ! ¢£ ¢ : : : : 85 random fraction of pop. that has HIV, then for person selected at If (1996 CDC) 476,999/242,200,000 = .00197 is the “specificity of test” “sensitivity of test” person does not have HIV person HAS HIV person tests negative person tests positive Suppose we know that ¢£ #¢ £  # ¢£ ¥ Rogers commission concluded disaster results unless ¤ EXAMPLE : (Like # 3.109, p. 158) : HIV test. EXAMPLE : Space shuttle Challenger disaster. The § ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 ¤ ! ¤ § ¤ © # © ¤ #© ¤ # % ¦ #¢ £ # 84 STA 2023 c B.Presnell & D.Wackerly - Lecture 7 ¢ !¥ ¤ ¢ ¥ ©¥ ¦ ¦ §% # ©¥ ¦ ©  %' ¡ ¡ Since Know ¢£ # ¡ Want ¤ ¤ ¥ ¦ #¢ £ ¡ , ¢£ ©#  ©¥  ¥ ¤ , ¢ Know ¢£ # ©¥ ¤ ¦ §%  ¦ ¦ : # ¤ #¢ £ #¢ £ ¥ #¢ £ ©¥ ¤ ¢£ , and (WHY?) # ¥ (cond.) prob. that the person actually has HIV ? ¡ #¢ £  ¥ ¤ ¦ ¢£ ¥ ¢£ ¥ ¤ ¥ ¤ ¤ A person is selected and tests positive. What is the ¦ % £ ¢ ¡ #¢ £ ¡ ¥ ¢£ ©#  ©¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 ¦ §'% ¥ £ % §% ¥ . 86 ¡ KNEW Thus ¦ ? #¢ £ # # ¡ ¦ ©  ¤ #£  ¤ . are mutually exclusive ¥ ¤  #¢ £ © ¢£ ¥ ¤ ? , found ¦ % §'%  ¤ and ¦ % §'% ¢£ #  ¥ ¤ ¦ % §'%  §% ¤ ¡ Note that How about ¢£ #¢ £ ¦ % §§ §   %% ¢  ¢ ¡ How about ¢£ #¢ £ # ¥ ¤ ¥ ¤ #¢ £ ¢£  ¥ ¤ ©¥ %' ¥ ¤ ¥ ¤ #£ ©  ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 ¡ ¤  ¥ ¦ §'%  ¥ ¤  ¦ %% §§ #¢ £ §  ¥ ¤  "§& #¢ £ ©¥ ¤ 87 ¡ according to rules of chance ( the “random” part) takes on different values (the “variable” part) Payoff or . – Toss a coin, bet $1 on “heads”, – last digit in social security number. – Number of phone calls to fire dept. in a day. – Number of accidents in G-ville next week. variables where the values change in “jumps”). different values of the variable is countable ( look for DISCRETE (Def. 4.2, p. 166)– the number of Two Types of Random Variables ¡ ¡ Intuitively : a random variable is a quantity that 88 Sample Point sum of up faces. Ex. Roll two dice. 36 sample points. Let sample point. that assigns one and only one numerical value to each Def. A RANDOM VARIABLE (Def. 4.1, p. 164) is a rule – Amount of rainfall (in inches) in a month – Height, Weight , Blood Pressure – Time to assemble automobile smoothly, without jumps) variables where the possible values change take on all values on a line interval (look for CONTINUOUS (Def. 4.3, p. 166) – the variable can STA 2023 c B.Presnell & D.Wackerly - Lecture 7 Chapter 4 : Discrete Random Variables § ¡ ¦ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 ¨§ §£ ¡ . . . ¡ ¡ & &£ "§ £ §¥ ¥ ¥ 12 3 . . . 2 89 ¡ % ¡ ¡ £ ¥ £ ¥ Note: Must have (grey shaded box, p. 169): value of the r.v. the prob., , associated with each possible Each possible value for the variable graph giving : distribution of a discrete r.v. is a formula, table, or ¡ ¡ £ ¥ (sum is over all values of ). for all poss. values ; random. : : 2nd camera selected is defective 1st camera selected is defective # defectives selected (0, 1, or 2). Ex. Shipment of 30 cameras, 6 defective. Select 2 at STA 2023 c B.Presnell & D.Wackerly - Lecture 7 § ¥£ § % ¥£ ¦ ¦ ¦ ¦" £ ¡ § ¦ '( % 90 ¥ ¢¤ £ ©¥ ¢£ " DEFINITION : (Def. 4.4, p. 169) The probability  ¢£ ¢¤ £ ¦ % ¦ ¢£ ¥ ¤ ¦ '( ¦ "' £¥ §¥  ¤  & £¥ ©¥ ¥ ¦" £ %¥ ¦ ¥ " ¦ ¥  ¥ ¢ ¤ £   ¢¤ £ % © ¢¤ £ ¥ ©   ¦ &' ©  ¥ " £¥ "( ¢¤ £ ¥ ¢£  ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 7  © ¦ ¥  ¤£  ¤©  ¥ © ¥ ¥ )( ¥ ¢¤ £ ©¥ ¢£ ©¥ 91 © ¤ %§  ¦ ''  '& £  ©¥ ¦ .6345 .3310 .0345 0 % ¥ 1 2 ¢£ ¥ ¢¤ £ ¥ "£ 92 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) (1,3), (2,2), (3,1) ... ... 4 5 6 ... ... ... (5,6), (6,5) (6,6) 8 ... (1,2), (2,1) 3 7 (1,1) Sample Points (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) 2 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) Ex. Sum of up faces on two dice. STA 2023 c B.Presnell & D.Wackerly - Lecture 7 ¦ ¢£ ¦ ¥ ") '& £ ¥ ¦" £¥ ¦ ¢¤ £ ¥ Table of the dist. of : ¤  ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 7  %' £ 9 10 11 12 (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 ¥ ¥ ¥ £ ¦ )( 93 0 1/36 2/36 3/36 4/36 5/36 6/36 1 p(x) 2 3 4 5 6 7 8 9 10 11 Distribution of the Sum of Two Fair Dice STA 2023 c B.Presnell & D.Wackerly - Lecture 7 12 x 94 ...
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This note was uploaded on 12/15/2011 for the course STA 2023 taught by Professor Ripol during the Spring '08 term at University of Florida.

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