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.