probability3

 due 9/24 9pm (MST)

 

typed answer in pdf required.

 

details required.

Math 464 – Fall

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3 – Homework 4

1. We roll a six-sided die n times. Each time the die comes up 1, we flip a
fair coin. Let X be the number of heads we get. Note that the number of
times the coin is flipped is random. Find the mean and variance of X.

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. You have n books on algebra, k on probability and l on calculus. The
books are all different. 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 adjacent and within each subject they are
in alphabetical order.

3. 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.

4. I have 15 identical cookies and 4 friends. I am going to give cookies to my
friends. In how many way can I do this if
(a) I give all the cookies away and there are no other constraints on how
many cookies each friend gets.
(b) I give all the cookies away and each friend gets at least one cookie.
(c) I don’t necessarily give away all the cookies and I do not require that
each friend get a cookie.

5. 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. )

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6. (Exposition:) I have 5 identical balls and I have three buckets which are
labelled A, B, C.
(a) In how many different ways can I put the five balls into the buckets.
(There are no constraints, e.g., a bucket can be empty.)
(b) In how many different ways can I put the five balls into the buckets if
each bucket must contain at least one ball?

Now suppose I perform the following experiment. For each ball I pick one of
the three buckets with equal probability and put the ball in that bucket. I
claim that the probability that each bucket has at least one ball is 50/81. One
might think this probability would be given by your answer to (b) divided
by your answer to (a), which should work out to 2/7.
(c) Explain why your answer to (b) divided by your answer to (a) is not the
correct probability.
(d) Derive my answer.

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