To determine [tex]\( P(B \mid A) \)[/tex], which is the conditional probability of event [tex]\( B \)[/tex] occurring given that event [tex]\( A \)[/tex] has occurred, we can use the formula for conditional probability:
[tex]\[
P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}
\][/tex]
Let's break it down step-by-step:
1. Identify the given probabilities:
- [tex]\( P(A) = 0.50 \)[/tex]
- [tex]\( P(B) = 0.80 \)[/tex]
- [tex]\( P(A \text{ and } B) = 0.20 \)[/tex]
2. Apply the conditional probability formula:
[tex]\[
P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}
\][/tex]
3. Substitute the known values into the formula:
[tex]\[
P(B \mid A) = \frac{0.20}{0.50}
\][/tex]
4. Perform the division:
[tex]\[
P(B \mid A) = \frac{0.20}{0.50} = 0.4
\][/tex]
Therefore,
[tex]\[
P(B \mid A) = 0.40
\][/tex]
The correct answer is:
A. 0.40
