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Chapter 7: Problem 47

Suppose the probability that Bill can solve a problem is \(p_{1}\) and the probability that Mike can solve it is \(p_{2}\). Show that the probability that Bill and Mike working independently can solve the problem is \(p_{1}+p_{2}-p_{1} p_{2}\).

Short Answer

Expert verified
The probability that Bill and Mike working independently can solve the problem is \(p_1 + p_2 - p_1 p_2\), which represents the probability that at least one of them solves the problem. This result is obtained by applying the formula for the probability of the union of two events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), where event A represents Bill solving the problem, event B represents Mike solving the problem, and \(P(A \cap B) = p_1 * p_2\).

Step by step solution

01

Identify the events and their probabilities

Event A (Bill solving the problem) has a probability of \(p_1\), while event B (Mike solving the problem) has a probability of \(p_2\). As they are working independently, the probability of both events happening at the same time (A and B, or A 鈭� B) is equal to \(p_1 * p_2\).
02

Apply the union formula

We are now going to use the formula for the probability of the union of two events: \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] Since \(P(A \cap B) = p_1 * p_2\), we can substitute it into the formula: \[P(A \cup B) = p_1 + p_2 - p_1 p_2\]
03

Interpret the result

The expression \(P(A \cup B)\) represents the probability that at least one of them solves the problem. So, the probability that Bill and Mike working independently can solve the problem is \(p_1 + p_2 - p_1 p_2\).

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