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Chapter 1: Problem 20

A person answers each of two multiple choice questions at random. If there are four possible choices on each question, what is the conditional probability that both answers are correct given that at least one is correct?

Short Answer

Expert verified
The conditional probability that both answers are correct given that at least one is correct is approximately 0.143

Step by step solution

01

Understand Conditional Probability

The formula for conditional probability is: \(P(A|B) = \frac{{P(A \cap B)}}{{P(B)}}\). In this question, we want to find the probability that both answers are correct (event A) given that at least one is correct (event B).
02

Calculate Probability of Each Event

Let's calculate the probability of each event. For a multiple choice question of four choices, the probability of getting an answer correct randomly is \( \frac{1}{4} = 0.25 \). So, \(P(A) = (0.25) * (0.25) = 0.0625\) which is the probability of both questions being answered correctly. \(P(B)\) which is probability of at least one question being answered correctly can be found out by using 1 - probability of both wrong = \(1 - (\frac{3}{4}) * (\frac{3}{4}) = 0.4375\)
03

Calculate Conditional Probability

Apply these values to the formula for conditional probability to get: \(P(A|B) = \frac{{0.0625}}{{0.4375}} = 0.14285714285714285.\)

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