Answer:
(b) What is the probability that exactly two of the three friends will sit next
to each other?
7E-10 You and two of your friends are in a group of 10 people. The group is
randomly split up into two groups of 5 people each. Specify an appropriate
sample space and determine the probability that you and your two friends
are in the same group.
7E-11 You are dealt a hand of four cards from a well-shuffled deck of 52
cards. Specify an appropriate sample space and determine the probability
that you receive the four cards J, Q, K, A in any order, with suit irrelevant.
7E-12 You draw at random five cards from a standard deck of 52 cards.
What is the probability that there is an ace among the five cards and a king
or queen?
7E-13 Three balls are randomly dropped into three boxes, where any ball is
equally likely to fall into each box. Specify an appropriate sample space and
determine the probability that exactly one box will be empty.
7E-14 An electronic system has four components labeled as 1, 2, 3, and 4.
The system has to be used during a given time period. The probability
that component i will fail during that time period is fi
for i = 1, . . . , 4.
Failures of the components are physically independent of each other. A
system failure occurs if component 1 fails or if at least two of the other
components fail. Specify an appropriate sample space and determine the
probability of a system failure.
7E-15 The Manhattan distance of a point (x, y) in the plane to the origin
(0, 0) is defined as |x| + |y|. You choose at random a point in the unit square
{(x, y) : 0 ≤ x, y ≤ 1}. What is the probability that the Manhattan distance
of this point to the point (0, 0) is no more than a for 0 ≤ a ≤ 2?
7E-16 You choose at random a point inside a rectangle whose sides have the
lengths 2 and 3. What is the probability that the distance of the point to the
closest side of the rectangle is no more than a given value a with 0 < a < 1?
7E-17 Pete tosses n + 1 fair coins and John tosses n fair coins. What is the
probability that Pete gets more heads than John? Answer this question first
for the cases n = 1 and n = 2 before solving the general case.
7E-18 Bill and Mark take turns picking a ball at random from a bag con-
taining four red balls and seven white balls. The balls are drawn out of the
bag without replacement and Mark is the first person to start. What is the
probability that Bill is the first person to pick a red ball?
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