By Sheldon Ross

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**Extra resources for A First Course In Probability (Solution Manual)**

**Sample text**

54. 2 Chapter 4 55. 2 e + e 2 2 56. The number of people in a random collection of size n that have the same birthday as yourself is approximately Poisson distributed with mean n/365. Hence, the probability that at least one person has the same birthday as you is approximately 1 − e−n/365. Now, e−x = 1/2 when x = log(2). Thus, 1 − e−n/365 ≥ 1/2 when n/365 ≥ log(2). That is, there must be at least 365 log(2) people. 57. (a) 1 − e−3 − 3e−3 − e−3 32 17 = 1 − e−3 2 2 17 1 − e−3 P{ X ≥ 3} 2 = (b) P{X ≥ 3X ≥ 1} = P{ X ≥ 1} 1 − e−3 59.

73. (a) 1/16, 74. Let PA be the probability that A wins when A rolls first, and let PB be the probability that B wins when B rolls first. Using that the sum of the dice is 9 with probability 1/9, we obtain upon conditioning on whether A rolls a 9 that PA = (b) 1/32, (c) 10/32, (d) 1/4, (e) 31/32. ) 75. (a) The probability that a family has 2 sons is 1/4; the probability that a family has exactly 1 son is 1/2. Therefore, on average, every four families will have one family with 2 sons and two families with 1 son.

2)3 97 10 = 10 . 8) 15 10 . 8) 2 33 10 . 8) 2 10 10 Ei are conditionally independent given the guilt or innocence of the defendant. 90. Let Ni denote the event that none of the trials result in outcome i, i = 1, 2. Then P(N1 ∪ N2) = P(N1) + P(N2) − P(N1N2) = (1 − p1)n + (1 − p2)n − (1 − p1 − p2)n Hence, the probability that both outcomes occur at least once is 1 − (1 − p1)n − (1 − p2)n + (p0)n. Chapter 3 39 Theoretical Exercises 1. P(ABA) = 2. If A ⊂ B P( AB ) P( AB ) ≥ = P(ABA ∪ B) P( A) P( A ∪ B) P( A) , P(ABc) = 0, P( B) P(AB) = 3.