To solve for $P(A \cap B)$, we can use the formula for the probability of the union of two events, which states:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Given values:

• $P(A) = \frac{6}{15}$
• $P(B) = \frac{8}{15}$
• $P(A \cup B) = \frac{4}{15}$

We substitute these values into the formula and solve for $P(A \cap B)$:

$\frac{4}{15} = \frac{6}{15} + \frac{8}{15} - P(A \cap B)$

Simplify by combining terms on the right:

$\frac{4}{15} = \frac{14}{15} - P(A \cap B)$

Now, isolate $P(A \cap B)$ by moving it to the other side:

$P(A \cap B) = \frac{14}{15} - \frac{4}{15}$

$P(A \cap B) = \frac{10}{15} = \frac{2}{3}$

So, the answer is:

$P(A \cap B) = \frac{2}{3}$

Would you like further details on this calculation?

Here are five related questions you may find helpful:

1. How would the answer change if $P(A \cup B)$ were different?
2. What is the interpretation of $P(A \cap B)$ in terms of overlap between events A and B?
3. How do we calculate $P(A \cup B \cup C)$ if a third event $C$ is introduced?
4. What is the probability of $A$ or $B$ not occurring?
5. How can we find $P(A | B)$ if given additional information?

Tip: When dealing with probability intersections, using the union formula is often helpful to find unknown probabilities when you have values for the union and individual probabilities.