Consider a situation in which [tex]$P(A)=\frac{1}{8}, P(C)=\frac{1}{4}$[/tex], and [tex]$P(A$[/tex] and [tex]$B)=\frac{1}{12}$[/tex]. What is [tex]$P(B$[/tex] and [tex]$C)$[/tex]?
Answer (2)
Use the independence of events A and B to find P ( B ) : P ( A and B ) = P ( A ) × P ( B ) .
Calculate P ( B ) : P ( B ) = 8 1 12 1 = 3 2 .
Use the independence of events B and C to find P ( B and C ) : P ( B and C ) = P ( B ) × P ( C ) .
Calculate P ( B and C ) : P ( B and C ) = 3 2 × 4 1 = 6 1 .
Explanation
Understand the problem and provided data We are given the probabilities P ( A ) = 8 1 , P ( C ) = 4 1 , and P ( A and B ) = 12 1 . We are also told that events A and B are independent, and events B and C are independent. Our goal is to find P ( B and C ) .
Use independence of A and B Since A and B are independent events, we know that P ( A and B ) = P ( A ) × P ( B ) . We can use this to find P ( B ) .
Solve for P(B) We have P ( A and B ) = P ( A ) × P ( B ) , so 12 1 = 8 1 × P ( B ) . To solve for P ( B ) , we divide both sides by 8 1 : P ( B ) = 8 1 12 1 = 12 1 × 1 8 = 12 8 = 3 2 .
Use independence of B and C Now that we have P ( B ) = 3 2 , we can use the fact that B and C are independent events to find P ( B and C ) . Since B and C are independent, P ( B and C ) = P ( B ) × P ( C ) .
Calculate P(B and C) We have P ( B ) = 3 2 and P ( C ) = 4 1 , so P ( B and C ) = P ( B ) × P ( C ) = 3 2 × 4 1 = 12 2 = 6 1 .
State the final answer Therefore, P ( B and C ) = 6 1 .
Examples
Understanding independent events is crucial in many real-world scenarios. For example, consider a quality control process in a factory where two machines operate independently. Machine A has a probability of 8 1 of producing a defective item, and the probability of both machines A and B producing a defective item is 12 1 . By calculating the probability of machine B producing a defective item, we can then determine the likelihood of both machine B and another independent process C (with a defect rate of 4 1 ) failing simultaneously, allowing for targeted improvements and risk management.
The probability P ( B and C ) is calculated to be 6 1 . This is derived using the independence of events B and C and the known probabilities of events A and B. First, we found P ( B ) which was 3 2 , then applied that to find P ( B and C ) .
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