CS 70 Secod Midter 7 April 2011 NAME (1 pt): SID (1 pt): TA (1 pt): Nae of Neighbor to your left (1 pt): Nae of Neighbor to your right (1 pt): Istructios: This is a closed book, closed calculator, closed coputer, closed etwork, ope brai exa, but you are peritted a 1 page, double-sided set of otes, large eough to read without a agifyig glass. You get oe poit each for fillig i the 5 lies at the top of this page. Each other questio is worth 20 poits. Write all your aswers o this exa. If you eed scratch paper, ask for it, write your ae o each sheet, ad attach it whe you tur it i (we have a stapler). 1 2 3 4 5 Total Aswers 1
Questio 1 (20 poits). For full credit explai your aswers. Part 1 (10 poits). Pick a rado iteger i the rage fro 0 to 9,999,999, each with equal probability. What is the probability that the decial digits of add up to 9? Fill i your aswer i the box below. P Aswer: We eed to cout the uber of ways you ca order 9 stars ad 6 bars, with the uber of stars betwee bars i ad i + 1 beig the value of decial digit i + 1: C(15, 6) 15!/(6!9!). (If the sequece starts (eds) with stars, these deterie the value of the first (last) decial digit.) The we divide by the uber of possible 7-digit ubers, 10 7, to get the aswer C(15, 6)/10 7. Part 2 (5 poits). What is the probability that the decial digits of add up to 10? Fill i your aswer i the box below. P Aswer: This is ot exactly stars-ad-bars with 10 stars ad 6 bars, because if all 10 stars ed up betwee two cosecutive bars, we ca t represet this as a sigle decial digit. So we eed to subtract out these C(7,1) 7 possibilities, yieldig (C(16, 6) 7)/10 7. Part 3 (5 poits). What is the probability that the decial digits of add up to 11? Fill i your aswer i the box below. P Aswer: Agai, this is ot exactly stars-ad-bars with 11 stars ad 6 bars, because we eed to subtract out the cases where oe digit (group of cosecutive stars) is 10 (ad oe other is 1) or oe digit is 11: The uber of ways oe digit could be 10 ad aother 1 is 7 6 42, ad the uber of ways oe digit could be 11 is 7, yieldig (C(17, 6) 42 7))/10 7. 2
Questio 1 (20 poits). For full credit explai your aswers. Part 1 (10 poits). Pick a rado iteger i the rage fro 0 to 99,999,999, each with equal probability. What is the probability that the decial digits of add up to 9? Fill i your aswer i the box below. P Aswer: We eed to cout the uber of ways you ca order 9 stars ad 7 bars, with the uber of stars betwee bars i ad i + 1 beig the value of decial digit i + 1: C(16, 7) 16!/(7!9!). (If the sequece starts (eds) with stars, these deterie the value of the first (last) decial digit.) The we divide by the uber of possible 8-digit ubers, 10 8, to get the aswer C(16, 7)/10 8. Part 2 (5 poits). What is the probability that the decial digits of add up to 10? Fill i your aswer i the box below. P Aswer: This is ot exactly stars-ad-bars with 10 stars ad 7 bars, because if all 10 stars ed up betwee two cosecutive bars, we ca t represet this as a sigle decial digit. So we eed to subtract out these C(8,1) 8 possibilities, yieldig (C(17, 7) 8)/10 8. Part 3 (5 poits). What is the probability that the decial digits of add up to 11? Fill i your aswer i the box below. P Aswer: Agai, this is ot exactly stars-ad-bars with 11 stars ad 7 bars, because we eed to subtract out the cases where oe digit (group of cosecutive stars) is 10 (ad oe other is 1) or oe digit is 11: The uber of ways oe digit could be 10 ad aother 1 is 8 7 56, ad the uber of ways oe digit could be 11 is 8, yieldig (C(18, 7) 56 8))/10 8. 3
Questio 2 (20 poits). For full credit explai your aswers. 1. (10 poits) Let the saple space Ω {0, 1, 2, 3}, ad let the probability of each saple poit be uifor. What is the probability of the evets A {1, 2}, B {2, 3}, C {1, 3}? Are evets A ad B idepedet? What is P r[a B C]? Aswer: P (A) P (B) P (C) 2 4 P (A B) 1 4 P (A)P (B), so they are idepedet P r[a B C] 2 3 2. (10 poits) Suppose you are give a bag cotaiig ubiased cois. You are told that 1 of these are oral cois, with heads o oe side ad tails o the other; however, the reaiig coi has heads o both its sides. Suppose you reach ito the bag, pick out a coi uiforly at rado, flip it ad get a head. What is the (coditioal) probability that this coi you chose is the fake (i.e., double-headed) coi? Aswer: Let F be the evet that the coi we picked fro the bag is fake, ad N is the evet that it is ot fake, ad let H be the evet that the coi we picked fro the bag coes up head. P (F H) P (F H) P (H) P (F )P (H F ) P (H F )+P (H N) P (F )P (H F ) P (F )P (H F )+P (N)P (H N) 1 1 + 1 1 2 Suppose you flip the coi k ties after pickig it (istead of just oce) ad see k heads. What is ow the coditioal probability that you picked the fake coi? Aswer: Let F be the evet that the coi we picked fro the bag is fake, ad N is the evet that it is ot fake, ad let H k be the evet that the coi we picked fro the bag coes up head k ties. P (F H k ) P (F Hk ) P (H k ) P (F )P (H k F ) P (H k F )+P (H k N) P (F )P (H k F ) P (F )P (H k F )+P (N)P (H k N) 1 1 + 1 1 2 k 4
Questio 2 (20 poits). For full credit explai your aswers. 1. (10 poits) Let the saple space S {a, b, c, d}, ad let the probability of each saple poit be uifor. What is the probability of the evets A {b, c}, B {c, d}, C {b, d}? Are evets A ad B idepedet? What is P r[a B C]? Aswer: P (A) P (B) P (C) 2 4 P (A B) 1 4 P (A)P (B), so they are idepedet P r[a B C] 2 3 2. (10 poits) Suppose you are give a bag cotaiig ubiased cois. You are told that 1 of these are oral cois, with heads o oe side ad tails o the other; however, the reaiig coi has tails o both its sides. Suppose you reach ito the bag, pick out a coi uiforly at rado, flip it ad get a tail. What is the (coditioal) probability that this coi you chose is the fake (i.e., double-tailed) coi? Aswer: Let F be the evet that the coi we picked fro the bag is fake, ad N is the evet that it is ot fake, ad let T be the evet that the coi we picked fro the bag coes up tails. P (F T ) P (F T ) P (T ) P (F )P (T F ) P (T F )+P (T N) P (F )P (T F ) P (F )P (T F )+P (N)P (T N) 1 1 + 1 1 2 Suppose you flip the coi s ties after pickig it (istead of just oce) ad see s tails. What is ow the coditioal probability that you picked the fake coi? Aswer: Let F be the evet that the coi we picked fro the bag is fake, ad N is the evet that it is ot fake, ad let T s be the evet that the coi we picked fro the bag coes up tails s ties. P (F T s ) P (F T s ) P (T s ) P (F )P (T s F ) P (T s F )+P (T s N) P (F )P (T s F ) P (F )P (T s F )+P (N)P (T s N) 1 1 + 1 1 2 s 5
Questio 3 (20 poits) Bayes Casio. At Bayes Casio i Las Vegas, there are two types of slot achies: Red ad Blue. Every achie of oe color results i a wi 10% of the tie, ad every achie of the other color results i a wi 25% of the tie. (A wi is whe the achie returs oey). Nobody kows which color wis ore frequetly, but you are 80% sure it s the Blue achies. You fid a Blue achie i the casio ad play a quarter. (a) 6 poits. Let $ be the evet the Blue achie wis. Let A be the evet that the Blue achie is a good (25%) oe. Write dow the followig probabilities: Aswer: P ($ A) 25% P ($ A) 75% P ($ A) 10% P ($ A) 90% P (A) 80% P (A) 20% (b) 7 poits. Suppose the Blue achie does ot wi. Give this evet ad your 80% iitial estiate, what is the probability that the Blue achies have the better wi rate (25%)? Feel free to write your aswer as a fractio. Show your work i order to ear ay partial credit. Aswer: Usig Bayes rule, the the total probability rule, the Bayes rule agai, we write P (A $) P ($) P ($ A) + P ($ A) + 30 39 77% (c) 7 poits. Repeat part (b), supposig istead that the Blue achie wis. Aswer: Siilar to part (b), we have P (A $) + 10 11 91% 6
Questio 3 (20 poits) Bayes Casio. At Bayes Casio i Las Vegas, there are two types of slot achies: Red ad Blue. Every achie of oe color results i a wi 15% of the tie, ad every achie of the other color results i a wi 20% of the tie. (A wi is whe the achie returs oey). Nobody kows which color wis ore frequetly, but you are 75% sure it s the Blue achies. You fid a Blue achie i the casio ad play a quarter. (a) 6 poits. Let $ be the evet the Blue achie wis. Let A be the evet that the Blue achie is a good (20%) oe. Write dow the followig probabilities: Aswer: P ($ A) 20% P ($ A) 80% P ($ A) 15% P ($ A) 85% P (A) 75% P (A) 25% (b) 7 poits. Suppose the Blue achie does ot wi. Give this evet ad your 75% iitial estiate, what is the probability that the Blue achies have the better wi rate (20%)? Feel free to write your aswer as a fractio. Show your work i order to ear ay partial credit. Aswer: Usig Bayes rule, the the total probability rule, the Bayes rule agai, we write P (A $) P ($) P ($ A) + P ($ A) + 48 65 74% (c) 7 poits. Repeat part (b), supposig istead that the Blue achie wis. Aswer: Siilar to part (b), we have P (A $) + 4 5 80% 7
Questio 4 ( 20 poits ) Bioial Distributio. 4.1 (10 poits). Suppose X is a rado variable that ca take positive iteger values 0, 1,..., ad β is soe real uber such that 0 β 1. Distributio of X is give by the followig recurrece relatio: { (1 β) for k 0 P (X k) β 1 β k+1 k P (X k 1) for k 1, 2,..., Use iductio to prove that X is actually a bioially distributed rado variable. What are the paraeters of the bioial distributio? What is E(X)? Aswer: We clai that X Bi(, β). I order to prove the clai, we have to show that the give recurrece relatio ca be siplified to the stadard for of the bioial distributio. I other words, we have to show that for all k 0, 1,..., We will use iductio to prove it. P (X k) ( k ) β k (1 β) k Base case (k 0): P (X 0) ( ) β 0 (1 β) 0 (1 β) Iductio Hypothesis: P (X k) ( ) k β k (1 β) k for all 0 k t, ad for soe t where 0 t <. As the iductio step, we ow have to show that ( ) P (X t + 1) β t+1 (1 β) (t+1) t + 1 Pluggig k t + 1 i the give recurrece relatio, we get P (X t + 1) β 1 β t 1 + 1 t + 1 ( β 1 β t t + 1 P (X t) ) β t (1 β) t [by iductio hypothesis] t β 1 β t t + 1! t!( t)! βt (1 β) t! (t + 1)!( t 1)! βt+1 (1 β) t 1 ( ) β t+1 (1 β) (t+1) t + 1 Hece, X Bi(, β), i.e. paraeters of the bioial distributio are ad β. Usig stadard result, we ca write E(X) β. 8
4.2 (10 poits). You have two boxes A ad B, each cotaiig balls. You radoly pick oe box ad the take oe ball out of it. You cotiue this process util you pick a box ad fid it epty. Suppose X is the uber of balls that reai i the other box whe you stop. If probability of pickig A ad B are p ad 1 p respectively, write dow the distributio of X i ters of ad p. Aswer: The evet {X k} eas that you ed up pickig a epty box ad the other box cotais k balls at that iteratio. There are two possible cases, viz. you pick box A ad fid it epty while box B cotais k balls, or you pick box B ad fid it epty while box A cotais k balls. X is, therefore, a iteger-valued rado variable that rages over {0, 1,..., }. Let A k deote the evet that you pick box A, fid it epty, but still there are k balls i box B. Siilarly, let B k deote the evet that you pick box B, fid it epty, but still there are k balls i box A. Therefore, we ca write P (X k) P (A k ) + P (B k ) Now, evet A k oly happes after you picked up exactly + k 2 k balls, all balls fro box A ad k balls fro box B i soe order, ad the pick box A agai (which is epty by ow) at the (2 k + 1)-th iteratio. Probability of takig out all balls fro box A ad k balls fro box B i 2 k iteratios is ( ) 2 k p (1 p) k ad probability of choosig box A i the (2 k + 1)-th iteratio is p. Therefore, ( ) 2 k P (A k ) p (1 p) k p ( ) 2 k p +1 (1 p) k Siilarly, P (B k ) ( ) 2 k p k (1 p) (1 p) ( ) 2 k p k (1 p) +1 Hece, the required distributio of X is (for all k 0, 1,..., ), P (X k) P (A k ) + P (B k ) ( ) ( 2 k 2 k p +1 (1 p) k + ( 2 k ) p k (1 p) +1 ) (p +1 (1 p) k + p k (1 p) +1 ) 9
Questio 4 ( 20 poits ) Bioial Distributio. 4.1 (10 poits). Suppose X is a rado variable that ca take positive iteger values 0, 1,..., ad µ is soe real uber such that 0 µ 1. Distributio of X is give by the followig recurrece relatio: { (1 µ) for k 0 P (X k) µ 1 µ k+1 k P (X k 1) for k 1, 2,..., Use iductio to prove that X is actually a bioially distributed rado variable. What are the paraeters of the bioial distributio? What is E(X)? Aswer: We clai that X Bi(, µ). I order to prove the clai, we have to show that the give recurrece relatio ca be siplified to the stadard for of the bioial distributio. I other words, we have to show that for all k 0, 1,..., ( ) P (X k) µ k (1 µ) k k We will use iductio to prove it. Base case (k 0): P (X 0) ( ) µ 0 (1 µ) 0 (1 µ) Iductio Hypothesis: P (X k) ( k) µ k (1 µ) k for all 0 k t, ad for soe t where 0 t <. As the iductio step, we ow have to show that ( ) P (X t + 1) µ t+1 (1 µ) (t+1) t + 1 Pluggig k t + 1 i the give recurrece relatio, we get P (X t + 1) µ 1 µ t 1 + 1 t + 1 µ 1 µ t t + 1 ( t P (X t) ) µ t (1 µ) t [by iductio hypothesis] µ 1 µ t t + 1! t!( t)! µt (1 µ) t! (t + 1)!( t 1)! µt+1 (1 µ) t 1 ( ) µ t+1 (1 µ) (t+1) t + 1 Hece, X Bi(, µ), i.e. paraeters of the bioial distributio are ad µ. Usig stadard result, we ca write E(X) µ. 10
4.2 (10 poits). You have two boxes R ad S, each cotaiig balls. You radoly pick oe box ad the take oe ball out of it. You cotiue this process util you pick a box ad fid it epty. Suppose Y is the uber of balls that reai i the other box whe you stop. If probability of pickig R ad S are q ad 1 q respectively, write dow the distributio of Y i ters of ad q. Aswer: The evet {Y k} eas that you ed up pickig a epty box ad the other box cotais k balls at that iteratio. There are two possible cases, viz. you pick box R ad fid it epty while box S cotais k balls, or you pick box S ad fid it epty while box R cotais k balls. Y is, therefore, a iteger-valued rado variable that rages over {0, 1,..., }. Let R k deote the evet that you pick box R, fid it epty, but still there are k balls i box S. Siilarly, let S k deote the evet that you pick box S, fid it epty, but still there are k balls i box R. Therefore, we ca write P (Y k) P (R k ) + P (S k ) Now, evet R k oly happes after you picked up exactly + k 2 k balls, all balls fro box R ad k balls fro box S i soe order, ad the pick box R agai (which is epty by ow) at the (2 k +1)-th iteratio. Probability of takig out all balls fro box R ad k balls fro box S i 2 k iteratios is ( ) 2 k q (1 q) k ad probability of choosig box R i the (2 k + 1)-th iteratio is q. Therefore, ( ) 2 k P (R k ) q (1 q) k q ( ) 2 k q +1 (1 q) k Siilarly, P (S k ) ( ) 2 k q k (1 q) (1 q) ( ) 2 k q k (1 q) +1 Hece, the required distributio of Y is (for all k 0, 1,..., ), P (Y k) P (R k ) + P (S k ) ( ) ( 2 k 2 k q +1 (1 q) k + ( 2 k ) q k (1 q) +1 ) (q +1 (1 q) k + q k (1 q) +1 ) 11
Questio 5 (20 poits) Rado Variables. Istead of a pair of the usual 6-sided dice, you ca play a gae with oe 4 sided die (sides ubered 1 through 4, each equally likely to coe up), ad oe 8 sided die (sides ubered 1 through 8, agai all equally likely). Let A be the su of the values that coe up o these two dice. 5.1 (5 poits) What is the expected value of A? E(A) Aswer: E(A) E(D4) + E(D8) E(D4) 4 i 1 4 1 4 i1 4 i i1 10 4 2.5 E(D8) 8 i 1 8 i1 8 1 8 i1 36 8 4.5 E(A) 2.5 + 4.5 E(A) 7 5.2 (5 poits) What is the probability A 8? P (A 8) Aswer: A 8 whe we roll either (1,7), (2,6), (3,5), or (4,4) P (A 8) 4 P (D4 x) P (D8 y) 4 1 4 1 8 4 32 1 8 12
5.3 (10 poits) You play a friedly bettig gae with your fried where you roll two dice each roud ad if the su of the two dice is 7 your fried pays you $5, otherwise you pay your fried $1. If you play this gae with two 6-sided dice your expected profit is $0 each roud. Defie a rado variable ad use it to calculate the expected aout of oey you wi or lose i 1 roud if you play usig a 4-sided die ad a 8-sided die. I the box below specify whether you wi or lose oey (by circlig the appropriate word) ad fill i the expected aout of oey you wi or lose i oe roud. I oe roud you expect to wi/lose (circle oe) $ Aswer: Let W be the aout of oey we wi. If we roll a 7 the W 5, otherwise W 1. There are 4 evets where A 7, we roll either (1,6), (2,5), (3,4), or (4,3). The probability A 7 is 4 36 1 8. The probability A is ay other uber is 7 8. E(W ) 5 P (A 7) 1 P (A! 7) 5 1 8 1 7 8 5 8 1 7 8 2 8 1 4 We expect to lose a quarter each roud. Note: For a pair of 6-sided dice P (A 7) 1 6 E(W ) 5 P (A 7) 1 P (A! 7) 5 1 6 1 5 6 5 6 1 5 6 0 13
Questio 5 (20 poits) Rado Variables. Istead of a pair of the usual 6-sided dice, you ca play a gae with oe 4 sided die (sides ubered 1 through 4, each equally likely to coe up), ad oe 8 sided die (sides ubered 1 through 8, agai all equally likely). Let S be the su of the values that coe up o these two dice. 5.1 (5 poits) What is the expected value of S? E(S) Aswer: E(S) E(D4) + E(D8) E(D4) 4 i 1 4 1 4 i1 4 i i1 10 4 2.5 E(D8) 8 i 1 8 i1 8 1 8 i1 36 8 4.5 E(S) 2.5 + 4.5 E(S) 7 5.2 (5 poits) What is the probability S 8? P (S 8) Aswer: S 8 whe we roll either (1,7), (2,6), (3,5), or (4,4) P (S 8) 4 P (D4 x) P (D8 y) 4 1 4 1 8 4 32 1 8 14
5.3 (10 poits) You play a friedly bettig gae with your fried where you roll two dice each roud ad if the su of the two dice is 7 your fried pays you $5, otherwise you pay your fried $1. If you play this gae with two 6-sided dice your expected profit is $0 each roud. Defie a rado variable ad use it to calculate the expected aout of oey you wi or lose i 1 roud if you play usig a 4-sided die ad a 8-sided die. I the box below specify whether you wi or lose oey (by circlig the appropriate word) ad fill i the expected aout of oey you wi or lose i oe roud. I oe roud you expect to wi/lose (circle oe) $ Aswer: Let W be the aout of oey we wi. If we roll a 7 the W 5, otherwise W 1. There are 4 evets where S 7, we roll either (1,6), (2,5), (3,4), or (4,3). The probability S 7 is 4 36 1 8. The probability S is ay other uber is 7 8. E(W ) 5 P (S 7) 1 P (S! 7) 5 1 8 1 7 8 5 8 1 7 8 2 8 1 4 We expect to lose a quarter each roud. Note: For a pair of 6-sided dice P (S 7) 1 6 E(W ) 5 P (S 7) 1 P (S! 7) 5 1 6 1 5 6 5 6 1 5 6 0 15