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Each of the members of a 7-judge panel independently makes

A First Course in Probability | 9th Edition | ISBN: 9780321794772 | Authors: Sheldon Ross ISBN: 9780321794772 63

Solution for problem 13STE Chapter 4

A First Course in Probability | 9th Edition

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A First Course in Probability | 9th Edition | ISBN: 9780321794772 | Authors: Sheldon Ross

A First Course in Probability | 9th Edition

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Problem 13STE

Problem 13STE

Each of the members of a 7-judge panel independently makes a correct decision with probability .7. If the panel’s decision is made by majority rule, what is the probability that the panel makes the correct decision? Given that 4 of the judges agreed, what is the probability that the panel made the correct decision?

Step-by-Step Solution:
Step 1 of 3

Solution 13STE

Step1 of 3:

Let us consider a random variable ‘X’ it presents the number of judges with correct decision.

Also, we have n = 7, and p = 0.7.

We need to find the probability that the panel makes the correct decision by using majority rule. And by using Baye’s Theorem.


Step2 of 3:

Here X follows a binomial distribution with parameters ‘n and p.’ the probability mass function of the binomial distribution is:

x = 0,1,2,...,n.

The majority of judges is agreed that correct decision, Then assume that judges 1,2, and 3 are not agreed and judges 4,5,6,and 7 agreed.

where,is obtained from Excel by using the function “=binomdist(X,n,p,false)”

X

P(X=4)

4

0.226895

5

0.317652

6

0.247063

7

0.082354

Total

0.873964

Therefore, P(Correct decision) = 0.8739 [by using majority rule].


Step3 of 3:

Consider Baye’s Theorem,

                                       

Where,

                                                                             

                                                                       

                                                     

Therefore,

                                   

                                                       

                                                 

           

                                                         

Therefore, P(4 Agreed) = 0.3242.

Now,

             

                 

                 

Therefore, The probability that the panel makes the correct decision is 0.70.


Step 2 of 3

Chapter 4, Problem 13STE is Solved
Step 3 of 3

Textbook: A First Course in Probability
Edition: 9
Author: Sheldon Ross
ISBN: 9780321794772

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Each of the members of a 7-judge panel independently makes