Solutions to Ma381, Exam 1 (version 2), F04

# Solutions to Ma381, Exam 1 (version 2), F04 - 1(18 pts In...

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Unformatted text preview: 1. (18 pts.) In 1912 the Titanic sank without enough lifeboats for the passengers and crew. The table below looks at survival by gender. Titanic Survival by Gmder , In working this problem, use the notation S = Survived the sinking D = did not survive the sinking (Died) M = Male ‘F = Female a. What proportion of the passengers were women? For full-credit, use correct probability notation when writing out your answer (i.e. write either P(AIB) or P(A) with A, B appropriately replaced by event(s) listed above). HF); 47" 2,214- m/m = ﬂ 2 .732 77» 0. Given that a passenger survived, what is the chance that they were female? For full- credit, use correct probability notation when writing out your answer (i.e. write either P(AIB) or P(A) with A, B appropriately replaced by event(s) listed above). P(Fl5)= 311’: '5 .4“ 7H 2. (12 pts.) Keno is a lottery game. Here are the details of a Keno bet you can make in Deadwood, South Dakota via video machine: You select 4 different numbers ﬁom 1 through 80 inclusive. There will be 20 numbers chosen as winners by the house, with the other 60 losers. Net winnings for a \$1 bet depend on how many of the 20 winning numbers you match. In particular, here are your winnings for the various possible numbers of matches: 7/0 (a “5mm {05"} aWhatisthechance,inasingle \$1 betathaiyouwinsgor? V/Muj‘n') -. P/ﬂ’dh'ki') : (”2402630) (if) b. What is the chance, in a single \$1 bet, that you lose a dollar? V/L'V—JI) = ﬂmm 0 0w rum, 1) 4‘5. V/rwmmh mem a.) z (15%?” , WM?) (3/ (:7 3. (12 pts.) Multiple Choice (circle the correct response, no work is nwded) a. If each of the three pairs of events AandB,AandC, &Bde have no outcomes in common, then we say m ii. A,B,C are independent events iii. A,B,C are complimentary events b. If P(A and B and C) = P(A)P(B)P(C) P(A and B) = P(A)P(B) P(A and C) = P(A)P(C) P(B and C) = P(B)P(C) then we say . A,B,C are inde . ndent ev . iii. A, B,C are complimentary events c. P(A) = P(A 1 B)P(B) + P(A | B')P(B') is < i. Alwais E ii. Sometimes true iii. >Never true d. P(A or B) = P(A) + P(B) is i. Always true a. - in. ‘ ever true 4. (24 pts.) A system consists of two components. The probability that the ﬁrst component works during its design life is 0.90, the probability that at least one of the two components does so is 0.95, and the probability that both components do so is 0.80. (Suggestion: Let F = ﬁrst component works during its design life, S = second component works during its design life, and draw a Venn—diagram.) +35 a. What is the probability that the second component works during its design life? P(5): , <95 b. Using the deﬁnition of independence, show why the two components do not work independently. 0.20 -- Focus) ¢P(F)P/§) (more) , = .765 c. What is the chance that exactly one component works (i.e. just the ﬁrst component works or just the second component works)? (((Fmvgﬁs‘) or (ﬂaw-15)) ’ -l5 d. Given the ﬁrst component works throughout its design life, what is the chance the second one does so? M) 0.90 ‘7 II‘ .87 5. (9 pts.) A system of n components is called a k—out-ofln system provided it functions if and only if at least k of its components function. (Example: Perhaps a roof being supported by n beams remains intact as long as at least some number of the beams are of a threshold level of strength.) a. A I-out-of-n system is better known as .series (circle one) system. _ b. A n-out-of-n system is better known as a parallecircle one) system. c. Suppose that the components in a 8-aut-oj110 system work independently with chance p = 0.9. Write, but don’t evaluate, an expression for the chance the system works. (’é’JCv’z-u‘ 4» (9’ch + (fikﬁ/‘ctf 6. (12 pts.) This problem concerns the tossing of a pair of four-sided (‘tetrahedral’) dice (each has the equally-likely numbers 1,2,3 ,4). a. If the pair are tossed once, what is the chance of a sum of 3? '2. '5 = ’— : .2 P/5UM 3) l6 l5 b. What is the chance of at least one double 4 in 10 tosses ofthe pair of dice? [Q’mr Wave: V9.25»: +) -: )- V020 Pwabﬁ 7") .ﬂ ’_ Kwoaoﬂt‘l- Ftny'jvl)’ L107... 324,2- ” ”OM 9' [041423) 3-,. /° ’3 I__(% 7. (12 pts.) In many popular board games (e. g. Trouble, Parchessi) one cannot move their players from “home” to the general playing area until they ﬁrst throw a particular value on the (single) die that is thrown. Suppose that a ‘6’ must be thrown, using a fair six- sided die, to bring a player out. a. What is the chance that you are ﬁrst able to bring out a player from home to the general playing area on your ﬁﬂh turn? (3%) b. What is the chance that it takes you at least three turns to bring out a player from home to the general playing area? For full-credit, give me a numeric answer. (”MT W 3mm”): /- Farm! ow 11w») 4:. I'CFCIMu)+ VCZ’IUNLSE 10:44“ 5 2 ' l-wa 27:5 = 3 8. (6 pts.) How many passwords of length 5 can be made where each character is either a lower case letter (‘a’,’b’,.. .’z’) or a digit (‘0’,’l’,...,’9’)? W’ W— 2‘ Kc. /o 47,2, / Isl»; 345 ...
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