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Find P(A | E) and P(C | F) using the tree diagram below. (Round your answers to four decimal places.)
P(A | E)	 	
 
P(C | F)	 	
 

To solve for \( P(A \mid E) \) and \( P(C \mid F) \) using the provided tree diagram, we will employ conditional probability and use the following formula:

\[
P(A \mid E) = \frac{P(A \cap E)}{P(E)}
\]
\[
P(C \mid F) = \frac{P(C \cap F)}{P(F)}
\]

Now, let's extract the necessary probabilities from the tree diagram.

### 1. \( P(A \cap E) \):
From the tree diagram, the probability of event A occurring followed by event E is calculated as follows:

\[
P(A \cap E) = P(A) \times P(E \mid A) = 0.2 \times 0.1 = 0.02
\]

### 2. \( P(B \cap E) \):
Similarly, the probability of event B followed by event E is:

\[
P(B \cap E) = P(B) \times P(E \mid B) = 0.2 \times 0.55 = 0.11
\]

### 3. \( P(C \cap E) \):
And the probability of event C followed by event E is:

\[
P(C \cap E) = P(C) \times P(E \mid C) = 0.6 \times 0.8 = 0.48
\]

### 4. \( P(E) \):
The total probability of event E is the sum of the above joint probabilities:

\[
P(E) = P(A \cap E) + P(B \cap E) + P(C \cap E) = 0.02 + 0.11 + 0.48 = 0.61
\]

### 5. \( P(A \mid E) \):
Using the formula for conditional probability:

\[
P(A \mid E) = \frac{P(A \cap E)}{P(E)} = \frac{0.02}{0.61} \approx 0.0328
\]

---

### 6. \( P(C \cap F) \):
Next, the probability of event C followed by event F:

\[
P(C \cap F) = P(C) \times P(F \mid C) = 0.6 \times 0.2 = 0.12
\]

### 7. \( P(F) \):
To find the total probability of event F, sum the joint probabilities involving F:

\[
P(F) = P(A \cap F) + P(B \cap F) + P(C \cap F)
\]

Where:
- \( P(A \cap F) = P(A) \times P(F \mid A) = 0.2 \times 0.9 = 0.18 \)
- \( P(B \cap F) = P(B) \times P(F \mid B) = 0.2 \times 0.45 = 0.09 \)
- \( P(C \cap F) = 0.12 \)

So:

\[
P(F) = 0.18 + 0.09 + 0.12 = 0.39
\]

### 8. \( P(C \mid F) \):
Finally, using the conditional probability formula:

\[
P(C \mid F) = \frac{P(C \cap F)}{P(F)} = \frac{0.12}{0.39} \approx 0.3077
\]

---

### Final Results:
\[
P(A \mid E) \approx 0.0328
\]
\[
P(C \mid F) \approx 0.3077
\]

Would you like further clarifications, or do you have any other questions? 

Here are 5 related questions for deeper exploration:
1. How do we compute \( P(B \mid E) \) from this tree diagram?
2. How would the probabilities change if we modified the branches from A to F?
3. What is the probability of \( P(B \mid F) \)?
4. How would you calculate the probability of \( P(E \cap F) \)?
5. How do these conditional probabilities relate to the concept of independence?

**Tip**: Conditional probabilities reflect the likelihood of an event occurring given the knowledge of another event's occurrence, and are key to understanding dependent events in probability theory.

Find P(A | E) and P(C | F) using the tree diagram below. (Round your answers to four decimal places.)
P(A | E)

P(C | F)

Learn how to calculate the conditional probabilities P(A | E) and P(C | F) from a probability tree diagram. This step-by-step solution covers joint probability, conditional probability formulas, and the conditional probability theorem, suitable for high school students (Grades 10-12).

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