To solve the problem, let's break it down step by step using the given information:

Problem:

We need to calculate $P(A \mid B)$, which is the probability of $A$ given $B$. This is defined as:
$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

Step 1: Identify the key values from the Venn diagram.

1. The number of items in $A \cap B$ (overlap of $A$ and $B$) is 3.
2. The total number of items in $B$ is the sum of the values in $B$: $P(B) = 3 + 5 = 8$

Step 2: Calculate $P(A \cap B)$ and $P(B)$.

The probability of $A \cap B$ is the number of items in $A \cap B$ divided by the total number of items in the universal set ($\xi$): $P(A \cap B) = \frac{3}{24}$ The total number of items in the universal set is: $9 + 7 + 3 + 5 = 24$

The probability of $B$ is: $P(B) = \frac{8}{24}$

Step 3: Calculate $P(A \mid B)$.

Using the formula $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$: $P(A \mid B) = \frac{\frac{3}{24}}{\frac{8}{24}} = \frac{3}{8}$

Final Answer:

$P(A \mid B) = \frac{3}{8}$

Would you like further details, or do you have any additional questions?

5 Related Questions:

1. How is conditional probability $P(A \mid B)$ different from $P(A \cap B)$?
2. Can we calculate $P(B \mid A)$ using this diagram?
3. What would the probability of selecting an item outside both $A$ and $B$ be?
4. How can we verify the total probability of all outcomes in the universal set?
5. What is the complement rule, and how does it apply to this problem?

Tip:

Always ensure the probabilities add up to 1 when summing across all mutually exclusive events in the universal set!