Compute the indicated probabilities by referring to the probability tree.
​(A) ​P(Mintersect​S)
​(B) ​P(R)
0.8
0.2
0.5
0.5
0.4
0.6
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Part 1
​(A) ​P(Mintersect​S)equals

enter your response here  ​(Type an integer or a​ decimal.)

To solve the problem, I'll explain how to calculate the probabilities \(P(M \cap S)\) and \(P(R)\) using the information provided and a probability tree structure.

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

Assuming that \( M \) and \( S \) are events in a probability tree, where the branches indicate conditional probabilities:

- Let's say the first branch splits into \( M \) and \( R \) with probabilities 0.8 and 0.2, respectively.
- Suppose after \( M \), there are two further branches representing \( S \) and \( \text{not } S \) with probabilities 0.5 and 0.5.
- After \( R \), the branches represent \( S \) and \( \text{not } S \) with probabilities 0.4 and 0.6.

Given this information, the probability of both \( M \) and \( S \) occurring is calculated by multiplying the probability along the path \( M \) followed by \( S \):

\[
P(M \cap S) = P(M) \times P(S \mid M)
\]

\[
P(M \cap S) = 0.8 \times 0.5 = 0.4
\]

So,

\[
P(M \cap S) = 0.4
\]

### Part B: \( P(R) \)

The probability \( P(R) \) is directly given as the probability of the branch corresponding to \( R \):

\[
P(R) = 0.2
\]

### Summary of Results:

- **(A)** \( P(M \cap S) = 0.4 \)
- **(B)** \( P(R) = 0.2 \)

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 probability \( P(\text{not } M \cup S) \)?
3. If \( P(M) \) were different, how would that affect \( P(M \cap S) \)?
4. Can you explain how the sum of \( P(M \cap S) \) and \( P(\text{not } M \cap S) \) relate to \( P(S) \)?
5. What is the probability of \( S \) given that \( R \) occurs?
6. How does the tree structure help in calculating conditional probabilities?
7. Can you describe the concept of independent events in this context?
8. What would change if the branches for \( S \) and \( \text{not } S \) were reversed?

**Tip:** When using a probability tree, ensure that the probabilities of all branches stemming from a single point add up to 1.

Learn how to compute probabilities using a probability tree with detailed explanations and examples. This problem involves calculating conditional probabilities and understanding the structure of a probability tree. Suitable for Grades 10-12 students.

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