Understanding Conditional Probability

The question asks us to find the value of \(P(B|A)\), which represents the conditional probability of event B occurring given that event A has already occurred. We are provided with the following probabilities:

• \(P(A) = 0.3\) (The probability of event A)
• \(P(B) = 0.4\) (The probability of event B)
• \(P(A|B) = 0.5\) (The probability of event A given event B)

To find \(P(B|A)\), we can use the definition of conditional probability and Bayes' Theorem.

Calculating the Intersection Probability \(P(A \cap B)\)

The formula for conditional probability \(P(A|B)\) is:

\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)

We can rearrange this formula to find the probability of the intersection of A and B, denoted as \(P(A \cap B)\):

\(P(A \cap B) = P(A|B) \times P(B)\)

Substituting the given values:

\(P(A \cap B) = 0.5 \times 0.4\)

\(P(A \cap B) = 0.20\)

So, the probability that both events A and B occur is 0.20.

Calculating Conditional Probability \(P(B|A)\) using Bayes' Theorem

Now, we can find \(P(B|A)\) using its definition:

\(P(B|A) = \frac{P(A \cap B)}{P(A)}\)

Substitute the values we have:

\(P(B|A) = \frac{0.20}{0.3}\)

\(P(B|A) = \frac{20}{30} = \frac{2}{3}\)

Converting the fraction to a decimal:

\(P(B|A) \approx 0.6667\)

Alternative Calculation using Bayes' Theorem Formula

Bayes' Theorem provides a direct way to calculate \(P(B|A)\) from the given information:

\(P(B|A) = \frac{P(A|B) \times P(B)}{P(A)}\)

Plugging in the provided values:

\(P(B|A) = \frac{0.5 \times 0.4}{0.3}\)

\(P(B|A) = \frac{0.20}{0.3}\)

\(P(B|A) = \frac{2}{3} \approx 0.6667\)

Conclusion

Both methods yield the same result. The calculated value for \(P(B|A)\) is \(\frac{2}{3}\), which is approximately 0.6667.