Let's solve the problem by analyzing the given probability tree and calculating the indicated probabilities.

Given Information:

The branches of the probability tree have the following values:

• $P(M) = 0.3$
• $P(\text{not } M) = 0.7$
• Conditional probabilities given $M$:
  • $P(S \mid M) = 0.9$
  • $P(\text{not } S \mid M) = 0.1$
• Conditional probabilities given $\text{not } M$:
  • $P(S \mid \text{not } M) = 0.7$
  • $P(\text{not } S \mid \text{not } M) = 0.3$

Part A: $P(M \cap S)$

To find $P(M \cap S)$, we multiply the probability of $M$ by the conditional probability of $S$ given $M$:

$P(M \cap S) = P(M) \times P(S \mid M)$

Substituting the values:

$P(M \cap S) = 0.3 \times 0.9 = 0.27$

So,

$P(M \cap S) = 0.27$

Part B: $P(R)$

In the problem statement, $R$ is not directly referenced, but if we assume $R$ represents the second major branch (i.e., $\text{not } M$):

$P(R) = P(\text{not } M) = 0.7$

Summary of Results:

• (A) $P(M \cap S) = 0.27$
• (B) $P(R) = 0.7$ (assuming $R = \text{not } M$)

If $R$ refers to a different event or if you have further details about it, please clarify.

Would you like more details, or do you have any questions?

Here are some related questions you might consider:

1. How would you calculate $P(\text{not } M \cap S)$?
2. What is the overall probability $P(S)$ based on the tree?
3. If $P(M)$ changes, how does that affect $P(M \cap S)$?
4. Can you describe the concept of mutually exclusive events in this context?
5. What is the probability of $S$ given that $\text{not } M$ occurs?
6. How can you verify the accuracy of your probability tree calculations?
7. What would be $P(\text{not } M \cup S)$?
8. How do dependent events affect the calculations in a probability tree?

Tip: Always ensure to double-check conditional probabilities and their respective paths when working with probability trees to avoid errors.