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STAT 394/ MATH 394 Probability I 2022
发布时间：2022-04-09
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Problem Set 2
STAT 394/ MATH 394
Probability I
2022
Question 1. [5 + 5 + 5 pts] Let Ak e A, k = 1, 2, . . ., be arbitrary events.
(a) Show that P ( 
Ak) < 
P (Ak). Hint: First prove it for A1, . . . , An, see Lemma 1.17. Then,
prove it for n → o using continuity of the probability measure.
(b) How useful is above inequality? Answer this question by considering the following two cases: (i) the Ak’s are such that P(Ak) = 3_k , and (ii) the Ak’s are such that P(Ak) = k_2 .
(c) Show that P ( 
Ak) = 1 if P(Ak) = k_2 for k = 1, 2, . . ..
Question 2. [5 pts] Suppose that you toss a fair coin repeatedly. Show that, with probability one, you will toss a head eventually. Hint: Introduce the events An = {“no head in the first n tosses”}, n = 1, 2, . . . .
Question 3. [5 + 5 pts] Consider tossing a fair coin n times. For k = 1, . . . , n, define the events Ak = {“the first k tosses yield only head”}.
(a) You want to model this experiment as a Laplace experiment. Find a suitable sample space and compute P(Ak).
(b) Describe in words the event Ak _1 \ Ak. What is the probability of this event?
Question 4. [5 + 5 pts] You roll a fair die 6 times.
(a) What is the probability that you roll at least one 6?
(b) What is the probability of rolling 5 different numbers?
Question 5. [5 + 5 pts] Consider an urn with 10 balls labeled 1, . . . , 10. You draw four times without replacement from this urn.
(a) What is the probability of only drawing balls with odd numbers?
(b) What is the probability that the smallest drawn number is equal to k for k = 1, . . . , 10?
Question 6. [5 pts] McMahon Hall on UW’s North Campus has 11 floors. You observe 7 people entering the elevator on the ground floor. In the absence of additional information you assume that every person is equally likely to leave the elevator on any floor. What is the probability that on each floor at most 1 person leaves the elevator?
Question 7. [0 + 0 + 0 pts] [Optional] Another (asymptotic) birthday problem.
(a) Show that the probability that one of the 121 students in MATH/ STAT 394 has the same birthday
as their professor is 1 - (1 - 1/365)121 .
(b) Let q(n, k) := 1 - (1 - 1/n)k. Show that for k = [αn] and α e (0, o),
lim q(n, k) = 1 - e_α .
-二
Remark: In fact, limn-二 q(n, k) = 0 for every fixed k > 0 and limk-二 q(n, k) = 1 for all fixed n 2 1. Choosing k = [αn] ensures that we obtain a non-trivial probability in the limit.
(c) The result in (b) differs markedly from the result obtained for the birthday problem discussed in the lecture. In plain English, what is the main difference between the two birthday problems? Can you make sense of how this difference results in the very different asymptotic behaviors?
