1. Suppose that the sample space S = {1, 2, 3, ...}. Let pk = Pr({k}) for k ∈ S. In each of the following cases, compute c.
(a) Suppose that pk = c(5/6)k for k ∈ S;
(b) Suppose that pk = c(5/6)k/(k)! for k ∈ S.
2. Suppose that the sample space is S = [0,∞). Let Bt = [t,∞) for any t ≥ 0. Suppose that Pr(Bt) = ce-6t for t ≥ 0. Compute (a)
c, (b) Pr(B2), and (c) Pr([1, 2)).
3. Suppose we are dealt 6 cards from a standard well-shuffled deck. What is the probability that there are
(a) a six card flush?
(b) 4 of one kind and 2 of another?
(c) two triples?
(d) 3 pairs?
(e) 2 pairs?
(f) 1 pair?
(g) at least one pair?
You may leave your answer in terms of n k .
4. Suppose that we have 30 different pairs of socks (60 socks in total) in a laundry basket. By different,we mean different colors or patterns so that each sock has a unique mate. Suppose that we select 10 socks at random from the basket. What is the Pr(Ak) for k = 0, . . . , 6 where Ak is the event that we have exactly k pairs among the 10 randomly selected socks? (Hint: To start, it might help to figure out which of those events have probability zero.)
5. Problem 1.5.3 on p. 54. 6. For the birthday problem let pk be the probability that k people all have different birthdays. In class, we derived an expression for pk assuming 366 days in a year that are equally likely to be someone's birthday. We can express pk+1 = pkfk where fk is a certain factor that depended on k.
(a) What is fk? (Hint: since pk is the probability that k people all have the distinct birthdays, we can think of fk as the conditional probability that one more person entering the room will also have a distinct birthday given that the k already in the room all have distinct birthdays. We are interested in computing 1 - pk for the number of people in the class. We are also interested in knowing when 1 - pk is greater than 1/2 since in these cases the bet is favorable for me. Set up an excel sheet as follows. In the first row, put the following headers: k, fk, pk, and 1 - pk. In the first column below k have the numbers from 1 up to 70. In the second column have the factor fk corresponding to the number k in the same row in the first column. In the first row, put the (obvious?) value for p1. The remainder of the pk's will be computed using the recursion; i.e., multiplying two numbers from the row above. Lastly, compute the last column with the values of 1 - pk.
(b) If there are 70 people in the class, what is the probability that I win the bet?
(c) What is the smallest class size where the probability of me winning is greater than 1/2?