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?

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.