Statistics 134: Concepts of Probability (Lugo)
Homework 5A
This assignment is due on Friday, October 7, in class. In order to do it you should read
Section 2.5.
The first exam is on Wednesday, September 28. You should at least attempt the recommended problems here before the exam. Among the required problems, Pitman 2.5.6 is good
as exam preparation, as is P17(a-b); P16 and P17(c) are probably too hard to be exam-type
problems.
There will also be a “homework 5B” due on Friday, October 7. Hand in the required
problems of both assignments 5A and 5B as a single “Homework 5” on October 7.
Recommended problems
From Pitman: 2.5.3, 2.5.7, 2.5.9, 2.5.12, 2.R.13, 2.R.15, 2.R.17, 2.R.23.
2.5.12 asks you to find the probabilities of various poker hands. This is a recommended problem, even though it’s even-numbered, because the answer is very easily found
in any number of references. See http://en.wikipedia.org/wiki/List_of_poker_hands
for more complete definitions of the hands and http://en.wikipedia.org/wiki/Poker_
probability for the solution.
(Remember: you don’t need to hand these in.)
Required problems
Pitman 2.5.6 In a hand of 13 cards drawn randomly from a pack of 52, find the chance of:
(a) no court cards (J, Q, K, A);
(b) at least one ace but no other court cards;
(c) at most one kind of court card.
P16: Playoff probabilities.
(a) I have 2n balls labeled 1, 2, . . . , 2n, and two boxes, East and West. I take n of the
balls at random (without replacement) and place them in the East box; I take the remaining
n balls and place them in the West box. I then take the smallest-numbered ball from each
box. What is the probability that I end up with balls 1 and 2?
For example, if n = 5 I might have 1, 3, 7, 8, 10 in East and 2, 4, 5, 6, 9 in West; then taking the
smallest-numbered ball from each box gives 1 and 2. On the other hand I could have 1, 2, 5, 6, 8 in
East and 3, 4, 7, 9, 10 in West, and then taking the smallest-numbered ball from each box gives 1
and 3.
(b) In the setup of (a), I take the two smallest-numbered balls from each box. What is
the probability that I end up with balls 1, 2, 3 and 4?
Both of the examples above satisfy this. On the other hand I might have 1, 3, 4, 8, 9 in one box
and 2, 5, 6, 7, 10 in the other; then taking the two smallest balls from each box gives 1, 2, 3, 5.
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(c) I have 3n balls labeled 1, 2, . . . , 3n, and three boxes, East, Central, and West. I take n
of the balls at random (without replacement) and place them in the East box; I take n of the
remaining 2n balls at random (without replacement) and place them in the Central box; I
take the remaining n balls and put them in the West box. I then take the smallest-numbered
ball from each box, and then the smallest-numbered ball among the balls remaining. What
is the probability that I end up with balls 1, 2, 3 and 4?
For example, if n = 4 I might have 1, 5, 8, 12 in East, 2, 6, 9, 10 in Central, and 3, 4, 7, 11 in
West. The smallest balls in each box are 1, 2, 3 and the smallest ball not among these is 4, so I
get the balls 1, 2, 3, 4. On the other hand I could have 3, 8, 9, 10 in East, 5, 6, 7, 11 in Central, and
1, 2, 4, 12 in West; then I get the balls 1, 2, 3, 5.
As you might imagine from the title of this problem and the nomenclature I’ve chosen,
this problem is intended to be a simple model of the probabilities that the teams with the best
regular-season records in a certain sports league make it to the post-season. In particular
Major League Baseball currently uses an arrangement very similar to (c), although there’s
a slight complication from the fact that the divisions in MLB are not all of the same size.
P17: Three-person birthday problem. In this problem, you will find an approximation of the probability that, among n people, three of them will have the same birthday.
(Note that in this problem we will make assumptions of independence that are not exactly
correct.)
(a) I pick three people at random. What is the probability that all three of them have
the same birthday (month and day)? Assume that all birthdays are equally likely and that
nobody is born on February 29. There are 365 days in a year.
(b) Say we have n people in our population. What is the expected number of triples of
people that have the same birthday?
(c) It is not true that “x, y, z share a birthday” and “u, v, w share a birthday” are independent whenever {x, y, z} =
6 {u, v, w}. Why not?
(d) Ignore (c), and assume that the events “x, y, z share a birthday” and “u, v, w share a
birthday” are independent whenever we have two different triplets. What is the approximate
probability that no three people have the same birthday, as a function of n? Use a Poisson
approximation.
(e) For what n is the function in (d) equal to 1/2?
You should consider whether your analysis here would work in the case of 2 people sharing the
same birthday. It turns out that the approximation made in (d) becomes increasingly bad as the
number of people intended to share a birthday grows.
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