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A state runs a lottery in which six numbers are randomly

Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger ISBN: 9781118539712 55

Solution for problem 148E Chapter 3.8

Applied Statistics and Probability for Engineers | 6th Edition

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Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger

Applied Statistics and Probability for Engineers | 6th Edition

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Problem 148E

A state runs a lottery in which six numbers are randomly selected from 40 without replacement. A player chooses six numbers before the state’s sample is selected.

(a) What is the probability that the six numbers chosen by a player match all six numbers in the state’s sample?

(b) What is the probability that five of the six numbers chosen by a player appear in the state’s sample?

(c) What is the probability that four of the six numbers chosen by a player appear in the state’s sample?

(d) If a player enters one lottery each week, what is the expected number of weeks until a player matches all six numbers in the state’s sample?

Step-by-Step Solution:

Step 1 of 5:

It is given that a state runs a lottery and 6 numbers are randomly selected from 40 numbers without replacement.

Also,it is given that a player chooses 6 numbers before the state selects the sample.

Using this we need to find the required values.

Step 2 of 5:

(a)

Here we have to find the probability that all the six numbers chosen by the player will match all the numbers selected by the state.

Let X be the numbers chosen .Then X follows the HyperGeometric distribution with parameters N=40,n=6,K=6.

The probability mass function of the HyperGeometric distribution is

P(X=x)=

Now we have to find the value of P(X=6).

P(X=6)=

        =

        =

        =

        =0.0000002605

Thus, the probability that all the 6 numbers chosen by the player matches the numbers chosen by the state is 0.0000002605.

Step 3 of 5:

(b)

Here we have to find the probability that 5 out of 6 numbers chosen by the player matches the numbers chosen by the state.

That is we have to find P(X=5) and it is given by

P(X=5)=

            =

            =

            =

            =

            =0.00005315

Thus, the probability that 5 of the 6 numbers chosen by the player matches the number chosen by the state is 0.00005315.

Step 4 of 5

Chapter 3.8, Problem 148E is Solved
Step 5 of 5

Textbook: Applied Statistics and Probability for Engineers
Edition: 6
Author: Douglas C. Montgomery, George C. Runger
ISBN: 9781118539712

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A state runs a lottery in which six numbers are randomly

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