Download Math 464 - Homework 3 1. We roll a die until we get a 1 and let N be

Math 464 - Homework 3
1. We roll a die until we get a 1 and let N be the number of rolls its takes
(including the final 1). Then we roll the die N more times and let X be the
number of 1’s that appear in this second set of N rolls.
(a) (easy) Find the expected value of N .
(b) (not so easy) Find the expected value of X. Hint: Condition on the
value of N .
• 2. A coin has probability p of heads. We flip a coin until we get n heads
in a row. Let X be the total number of flips. Find the expected value of X.
Hint: The idea is to use the partition theorem and the fact that whenever
you get a T, things start over. Consider the following finite partition. B1 is
the event that the first flip is T. B2 is the event that the first two flips are
HT. B3 is the event that the first three flips are HHT, and so on until Bn is
the event that the first n − 1 flips are H and then the nth flip is T. Finally
let B0 be the event that the first n flips are H.
3. You have a books on algebra, b on probability and c on calculus. If you
place them on a shelf at random what is the probability that
(a) Books on the same subject are adjacent.
(b) Books on the same subject are in alphabetical order by author, but not
necessarily adjacent.
(c) Books on the same subject are in alphabetical order by author and adjacent.
4. The usual deck of cards has 52 cards. There are 4 suits (hearts, diamonds,
clubs and spades) and each suit contains 13 ranks (ace, 2, 3, 4, 5, 6, 7, 8, 9,
10, jack, queen, king). You are dealt five cards. Find the probability of
(a) “four of a kind.” This means four cards of the same rank.
(b) a “full house.” This means three cards with the same rank and two cards
with the same rank. For example three queens and two 4’s.
(c) “three of a kind.” This means three cards with the same rank but you
do not have a full house.
5. An urn contains 4n balls. The balls are either red, yellow, green or blue.
There are n of each color. We draw r balls without replacement with r ≥ 4.
Find the probabilities that
(a) At least one ball is blue.
(b) Exactly two balls are blue.
(c) There is at least one ball of each color.
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6. Using the fact that
elements, show that
n
k
is the number of k element subsets of a set with n
k X
m
n
m+n
=
j
k−j
k
j=0
I am not looking for an algebraic derivation of this identity.
7. A round table has n seats. n people are seated at random around the
table. Fred dislikes two of the people. Let X be the number of neighbors
of Fred whom he dislikes. Find the p.m.f. of X. (Note that X can only be
0, 1, 2. )
• 8. 5 friends go to the movies. There are three movies showing. If all I care
about is how many friends go to each movie, not which particular friends go
to which movie,
(a) in how many ways can they buy tickets?
(b) in how many ways can they buy tickets if each movie must be seen by at
least one friend?
Now suppose that each friend picks a movie to see at random and their
choices are independent. I claim that the probability that each movie is seen
by at least one friend is 150/35 .
(c) Explain why your answer to (b) divided by your answer to (a) is not the
correct probability.
(d) Derive my answer.
• 9. (More ruined gamblers) We recall the gamblers ruin problem. Player
A start with a coins and player B with b coins. When A loses the game,
she gives a coin to player B. When B loses the game, he gives a coin to
player A. The probability that A wins a single game is p. They stop playing
when one of them has no coins left. The player with all a + b coins at the
end is obviously the winner. The goal of this problem is to compute their
probabilities of winning. As in class, we consider multiple experiments. In
experiment k, player A starts with k coins and player B with a + b − k coins.
(In all the experiments and at all times there are a total of a + b coins.) Let
pk be the probability that A wins in experiment k.
(a) Show that
pk = ppk+1 + (1 − p)pk−1 ,
k = 1, 2, · · · a + b − 1
(b) What are p0 and pa+b ?
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(c) Show that if p = 1/2, then the solution of this system of linear equations
is
pk =
k
a+b
You do not have to solve the equations, just check that this is indeed a
solution.
(d) Show that if p 6= 1/2 the solution is
pk =
rk − 1
,
r a+b − 1
r = (1 − p)/p
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