https://file.aiquickdraw.com/mathbot/file/1726896879873489xrvz7.jpg

The given image displays a probability tree diagram. Let's go over the steps to solve the questions based on this tree diagram.

### Given:
- From the first node, the branches represent:
  - \( P(A) = \frac{2}{3} \)
  - \( P(A') = \frac{1}{3} \)
  
- From node \( A \), the next set of branches represent:
  - \( P(B | A) = \frac{3}{4} \)
  - \( P(B' | A) = \frac{1}{4} \)
  
- From node \( A' \), the next set of branches represent:
  - \( P(B | A') = \frac{3}{8} \)
  - \( P(B' | A') = \frac{5}{8} \)

Now, let's answer the given questions:

### (a) Find \( P(B' | A') \)

This is directly given in the tree:
\[
P(B' | A') = \frac{5}{8}
\]

### (b) Find \( P(B' \cap A') \)

This is the joint probability of both events \( B' \) and \( A' \). According to the tree, the joint probability is:
\[
P(B' \cap A') = P(A') \times P(B' | A')
\]
Substitute the values:
\[
P(B' \cap A') = \frac{1}{3} \times \frac{5}{8} = \frac{5}{24}
\]

### Final Answers:
- (a) \( P(B' | A') = \frac{5}{8} \)
- (b) \( P(B' \cap A') = \frac{5}{24} \)

Let me know if you want further clarification or more details.

Here are some related questions to practice:
1. What is \( P(A \cap B) \)?
2. How would you calculate \( P(B | A) \)?
3. What is the total probability of \( P(A') \)?
4. How does the probability tree help visualize conditional probabilities?
5. What is the sum of the probabilities of all branches from \( A' \)?

**Tip:** Always remember that the sum of probabilities from any node in a probability tree must equal 1.

Find P(B' | A') and P(B' ∩ A') using the probability tree diagram.

Learn how to calculate conditional and joint probabilities using a probability tree. This example explains how to find P(B' | A') and P(B' ∩ A') through visualizing the tree structure and applying key probability formulas. Ideal for students in Grades 9-12.

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