Let *P*(*E*) = 0.25, *P*(*F*) = 0.35, and *P*(*F* ∩ *E*) = 0.1. Draw a Venn diagram and find the conditional probabilities.

(a)    *P*(*E* | *F*C )

(b)    *P*(*F* | *E*C )

Here is the Venn diagram that represents the events \(E\) and \(F\) with the given probabilities. Now, let's calculate the conditional probabilities.

### (a) \(P(E \mid F^C)\)
Using the formula for conditional probability:
\[
P(E \mid F^C) = \frac{P(E \cap F^C)}{P(F^C)}
\]
We know:
- \( P(F^C) = 1 - P(F) = 1 - 0.35 = 0.65 \)
- \( P(E \cap F^C) = P(E) - P(E \cap F) = 0.25 - 0.1 = 0.15 \)

Thus:
\[
P(E \mid F^C) = \frac{0.15}{0.65}
\]

### (b) \(P(F \mid E^C)\)
Similarly:
\[
P(F \mid E^C) = \frac{P(F \cap E^C)}{P(E^C)}
\]
We know:
- \( P(E^C) = 1 - P(E) = 1 - 0.25 = 0.75 \)
- \( P(F \cap E^C) = P(F) - P(F \cap E) = 0.35 - 0.1 = 0.25 \)

Thus:
\[
P(F \mid E^C) = \frac{0.25}{0.75}
\]

Now, let's compute the exact values.The conditional probabilities are:

- (a) \( P(E \mid F^C) = 0.231 \)
- (b) \( P(F \mid E^C) = 0.333 \)

Would you like more details or have any questions?

Here are 5 related questions:
1. How is the complement of an event used in conditional probability?
2. What is the relationship between joint probability and conditional probability?
3. How would the results change if \( P(E \cap F) \) were different?
4. How can you extend this to more than two events?
5. What is the significance of independent events in conditional probability?

**Tip:** Conditional probability helps to understand the likelihood of an event happening given the occurrence or non-occurrence of another event.

Let P(E) = 0.25, P(F) = 0.35, and P(F ∩ E) = 0.1. Draw a Venn diagram and find the conditional probabilities. (a) P(E | F^C), (b) P(F | E^C)

Explore how to calculate the conditional probabilities of P(E | F^C) and P(F | E^C) given P(E) = 0.25, P(F) = 0.35, and P(F ∩ E) = 0.1. This problem covers key concepts in probability theory, including complementary events and set theory, suitable for Grades 10-12.

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