From the sample space $S=\{1,2,3,4, \ldots, 15\}$, a single number is to be selected at random. Given the following events, find the indicated probability. A: The selected number is even. B. The selected number is a multiple of 4 . C. The selected number is a prime number \[ \mathrm{P}(\mathrm{B} \mid \mathrm{A}) \] \[ \mathrm{P}(\mathrm{B} \mid \mathrm{A})=\square \] (Simplify your answer.)

Step 1 ：The problem is asking for the conditional probability of event B (the selected number is a multiple of 4) given that event A (the selected number is even) has occurred.

Step 2 ：To find this, we need to know the total number of outcomes in event A and the number of outcomes in event B that are also in event A.

Step 3 ：Event A consists of all even numbers in the sample space, which are {2, 4, 6, 8, 10, 12, 14}. So there are 7 outcomes in event A.

Step 4 ：Event B consists of all multiples of 4 in the sample space, which are {4, 8, 12}. So there are 3 outcomes in event B.

Step 5 ：The outcomes in event B that are also in event A are still {4, 8, 12}, so there are 3 such outcomes.

Step 6 ：The conditional probability P(B|A) is then the number of outcomes in B that are also in A divided by the total number of outcomes in A. This can be calculated as follows: \( \frac{3}{7} \approx 0.43 \)

Step 7 ：Final Answer: The conditional probability P(B|A) is approximately \(\boxed{0.43}\).