A and B will take the same 10-question examination. Each

Chapter 4, Problem 42P

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QUESTION:

\(A\) and \(B\) will take the same 10 -question examination. Each question will be answered correctly by \(A\) with probability \(.7\), independently of her results on other questions. Each question will be answered correctly by \(B\) with probability \(.4\), independently both of her results on the other questions and on the performance of \(A\).

(a) Find the expected number of questions that are answered correctly by both \(A\) and \(B\).

(b) Find the variance of the number of questions that are answered correctly by either \(A\) or \(B\).

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QUESTION:

\(A\) and \(B\) will take the same 10 -question examination. Each question will be answered correctly by \(A\) with probability \(.7\), independently of her results on other questions. Each question will be answered correctly by \(B\) with probability \(.4\), independently both of her results on the other questions and on the performance of \(A\).

(a) Find the expected number of questions that are answered correctly by both \(A\) and \(B\).

(b) Find the variance of the number of questions that are answered correctly by either \(A\) or \(B\).

ANSWER:

Step 1 of 2

(a)

We are asked to find the expected number of questions that are answered correctly by both  and

 and  will take the same  question examination.

We have given the probability of  answering one question correctly,

The probability of  answering one question correctly,

Let  denote the number of questions that are answered correctly.

We need to find

Hence  follows a binomial random variable, we know a binomial distribution

………….(1)

The expected value of a binomial random variable is,

The probability that one question is answered correctly by both  and  is

Using independence rule of probability,

A binomial random variable with parameters

Hence the expected value is

Hence the expected number of questions that are answered correctly by both  and  is approximately

 

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