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Using the Venn Diagram below, what is the conditional probability of event Q occurring, assuming that event P has already happened [P(Q|P)]?

0.55

0.73

0.05

0.75

To calculate the conditional probability \( P(Q|P) \), we use the formula for conditional probability:

\[
P(Q|P) = \frac{P(Q \cap P)}{P(P)}
\]

From the Venn diagram:

- \( P(Q \cap P) = 0.4 \) (the intersection of \( P \) and \( Q \)),
- \( P(P) = 0.15 + 0.4 = 0.55 \) (the total probability of event \( P \)).

Now, calculate \( P(Q|P) \):

\[
P(Q|P) = \frac{0.4}{0.55} \approx 0.727
\]

So, the closest answer is \( 0.73 \).

Would you like more details or have any questions?

Here are 5 follow-up questions to consider:
1. How would you calculate \( P(P|Q) \) given the diagram?
2. What is the probability of \( P \cup Q \) (the union of \( P \) and \( Q \))?
3. How does the complement rule apply to this problem?
4. What is the interpretation of conditional probability in this context?
5. How would you calculate the independent probability of both events?

**Tip**: In Venn diagrams, remember to sum all areas relevant to the set to get total probabilities.

Using the Venn Diagram below, what is the conditional probability of event Q occurring, assuming that event P has already happened [P(Q|P)]?

Explore how to calculate the conditional probability of event Q, assuming event P has occurred, using the Venn Diagram. This example uses the formula P(Q|P) = P(Q ∩ P) / P(P) and covers probability concepts suitable for students in Grades 9-12.

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