INDE 303: Operations Research II
Homework Assignment 1
Due Date: February 13, 2020 (In class)
Chapter 1
Problem 26
A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards
each. Define events E1 , E2 , E3 , and E4 as follows:
E1
E2
E3
E4
= {the
= {the
= {the
= {the
first pile has exactly 1 ace},
second pile has exactly 1 ace},
third pile has exactly 1 ace},
fourth pile has exactly 1 ace}.
Find IP{E1 E2 E3 E4 }.
Problem 39
Stores A, B, and C have 50, 75, and 100 employees, and, respectively, 50, 60, and 70 percent of
these are women. Resignations are equally likely among all employees, regardless of sex. One
employee resigns and this is a woman. What is the probability that she works in store C?
Chapter 2
Problem 25-26
Suppose that two teams are playing a series of games, each of which is independently won by
team A with probability p and by team B with probability 1 − p. The winner of the series is the
first team to win i games.
1. If i = 4, find the probability that a total of 7 games are played. Also show that this
probability is maximized when p = 1/2.
2. Find the expected number of games that are played when i = 2.
3. Find the expected number of games that are played when i = 3.
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Problem 37
Let X1 , X2 , . . . , Xn be independent random variables, each having a uniform distribution over
(0, 1). Let M = maximum(X1 , X2 , . . . , Xn ). Show that the distribution function of M , FM (·),
is given by
FM (x) = xn , 0 ≤ x ≤ 1.
Problem 41
Consider n independent flips of a coin having probability p of landing heads. Say a changeover
occurs whenever an outcome differs from the one preceding it. For instance, if the results of
the flips are HHT HT HHT , then there are a total of five changeovers. Compute the expected
number of changeovers.
Problem 60
Calculate the moment generating function of the uniform distribution on (0, 1). Obtain IE{X}
and Var{X} by differentiating.
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