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Chapter 5: Q 5.155. (page 242)
Give two examples of Bernoulli trials other than those presented in the text.
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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).
Dice. When one die is rolled, the following six outcome are possible;
List the outcome constituting
A = event the die come up even
B = event the die come up 4 or more
C event the die come up at most 2, and
D= event the die come up 3.
ASU Enrollment Summary. According to the Arizona State University Enrollment Summary, a frequency distribution for the number of undergraduate students attending Arizona State University ( ASU ) in the Fall 2012 semester, by class level, is as shown in the following table. Here , 1 = freshman , 2 = sophomore , 3 = junior , and 4 = senior.
| Class level | 1 | 2 | 3 | 4 |
| No. of students | 9,652 | 11,115 | 17,302 | 21,114 |
Let X denote the class level of a randomly selected ASU undergraduate.
(a) What are the possible values of the random variable X?
(b) Use random-variable notation to represent the event that the student selected is a junior ( class - level 3 ).
(c) Determine P ( X = 3 ), and interpret your answer in terms of percentages.
(d) Determine the probability distribution of the random variable X.
(e) Construct a probability histogram for the random variable X.
The probability is 0.667 that the favorite in a horse race will finish in the money (first, second, or third place). In 500 horse races, roughly how many times will the favorite finish in the money?
Die and coin. Consider the following random experiment : First , roll a die and observe the number of dots facing up: then toss a coin the number of times that the die shows and observe the total number of heads. Thus , if the die shows three dots facing up and the coin (which is then tossed tree times) comes up heads exactly twice, then the outcome of the experiment can be represent as (3,2).
Part (a) Determine a sample space for this experiment.
Part (b) Determine the events that the total number of heads is even.
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