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Chapter 5: Q 5.86. (page 216)
Suppose that A and B are events such that ,and
Part (a). Are event A and B mutually exclusive ? Explain your answer.
Part (b) Find
Part (a) and are not mutually exclusive because and are not zero.
Part (b)
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The Hypergeometric Distribution. In this exercise, we discuss the hypergeometric distribution in more detail. When sampling is done without replacement from a finite population, the hypergeometric distribution is the exact probability distribution for the number of members sampled that have a specified attribute. The hypergeometric probability formula is
,
where Xdenotes the number of members sampled that have the specified attribute, Nis the population size, nis the sample size, and pis the population proportion.
To illustrate, suppose that a customer purchases 4 fuses from a shipment of 250, of which 94 % are not defective. Let a success correspond to a fuse that is not defective.
(a) Determine N, n, and p.
(b) Apply the hypergeometric probability formula to determine the probability distribution of the number of nondefective fuses that the customer gets.
Key Fact 5.6 shows that a hypergeometric distribution can be approximated by a binomial distribution, provided the sample size does not exceed 5% of the population size. In particular, you can use the binomial probability formula
with , to approximate the probability distribution of the number of nondefective fuses that the customer gets.
(c) Obtain the binomial distribution with parameters .
(d) Compare the hypergeometric distribution that you obtained in part (b) with the binomial distribution that you obtained in part (c).
An experiment has 40 possible outcomes, all equally likely. An event can occur in 25 ways. The probability that the event is .
Dice. Refer to exercise 5.53.
a Are events A and B mutually exclusive?
b Are events B and C mutually exclusive?
c Are events A, C and D mutually exclusive?
d Are there three mutually exclusive events among A, B, C and D? four?
Craps. The game of craps is played by rolling two balanced dice. A first roll of a sum of 7 or 11 wins; and a first roll of a sum of 2,3 , or 12 loses. To win with any other first sum, that sum must be repeated before a sum of 7 is thrown. It can be shown that the probability is 0.493 that a player wins a game of craps. Suppose we consider a win by a player to be a success,
a. Identify the success probability,
b. Construct a table showing the possible win-lose results and their probabilities for three games of craps. Round each probability to three decimal places.
c. Draw a tree diagram for part (b).
d. List the outcomes in which the player wins exactly two out of three times.
e. Determine the probability of each of the outcomes in part (d). Explain why those probabilities are equal.
f. Find the probability that the player wins exactly two out of three times.
g. Without using the binomial probability formula, obtain the probability distribution of the random variable , the number of times out of three that the player wins.
h. Identify the probability distribution in part (g).
Normality Requirement What is different about the normality requirement for a confidenceinterval estimate of \(\sigma \)and the normality requirement for a confidence interval estimateof \(\mu \)?
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