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A and B will take the same 10-question examination. Each
Chapter 4, Problem 42P(choose chapter or problem)
\(A\) 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 \(B\).
Questions & Answers
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