To find the conditional probability [tex]\( P(B \mid C) \)[/tex], we need to determine the probability that event B occurs given that event C has already occurred.
Here is a step-by-step breakdown of how to solve for [tex]\( P(B \mid C) \)[/tex]:
1. Identify the relevant events from the table:
- Event B: characterized by row 'B' in the table.
- Event C: characterized by column 'C' in the table.
2. Find the value of [tex]\( P(B \cap C) \)[/tex]:
- This is the number of times both B and C occur together. Looking at the table, we can see that the count for both B and C is 9.
3. Find the total number of occurrences of event C:
- This is given in the total column under C, which is 24.
4. Calculate the conditional probability [tex]\( P(B \mid C) \)[/tex] using the formula:
[tex]\[
P(B \mid C) = \frac{P(B \cap C)}{P(C)}
\][/tex]
Substituting the values we found:
[tex]\[
P(B \mid C) = \frac{9}{24}
\][/tex]
5. Simplify the fraction and convert to a decimal:
[tex]\[
\frac{9}{24} = \frac{3}{8} \approx 0.375
\][/tex]
6. Round the result to two decimal places:
[tex]\[
P(B \mid C) \approx 0.38
\][/tex]
So, the conditional probability [tex]\( P(B \mid C) \)[/tex] is indeed [tex]\( \frac{9}{24} = 0.38 \)[/tex].
Thus, the correct answer is:
[tex]\[ P(B \mid C) = \frac{9}{24} = 0.38 \][/tex]
