Compute the indicated probabilities by referring to the probability tree.

​(A)****

​P(Mintersect∩​S)

​(B)**** ​P(R)

0.30.70.40.60.60.4

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Part 1

​(A)****

​P(Mintersect∩​S)equals=0.180.18

​(Type an integer or a​ decimal.)

Part 2

​(B)****

​P(R)equals=enter your response here

​(Type an integer or a​ decimal.)

To compute the probabilities using the probability tree, let's assume you have a tree with events \(M\), \(S\), and \(R\) with the corresponding probabilities given. 

Let's break down the problem:

### Part A: \( P(M \cap S) \)
You've already calculated \( P(M \cap S) = 0.18 \). This is the probability that both events \(M\) and \(S\) happen simultaneously. 

### Part B: \( P(R) \)
To find the probability of \(R\), \( P(R) \), we need to look at all the branches that lead to event \(R\) and sum their probabilities.

Without specific details from the tree diagram, I will assume the general structure as follows:

- There are paths leading to \(R\) with certain probabilities.
- These paths may involve intermediate events, such as \(M\) and \(S\), and we multiply the probabilities along the paths to get the joint probabilities, then sum them to find \( P(R) \).

Let's say you have two paths leading to \(R\) as an example:

1. Path 1: Probability \(P(M) \times P(S \mid M) \times P(R \mid S)\)
2. Path 2: Probability \(P(\text{not } M) \times P(S \mid \text{not } M) \times P(R \mid S)\)

To compute \( P(R) \), sum the probabilities from each path:

\[
P(R) = P(M) \times P(S \mid M) \times P(R \mid S) + P(\text{not } M) \times P(S \mid \text{not } M) \times P(R \mid S)
\]

Given values:

1. \( P(M) = 0.3 \) and \( P(\text{not } M) = 0.7 \)
2. Let's assume \( P(S \mid M) = 0.6 \), \( P(S \mid \text{not } M) = 0.4 \)
3. Assume \( P(R \mid S) = 0.4 \)

So the probability \( P(R) \) would be:

\[
P(R) = 0.3 \times 0.6 \times 0.4 + 0.7 \times 0.4 \times 0.4
\]

Now let's calculate that:

\[
P(R) = 0.072 + 0.112 = 0.184
\]

So, \( P(R) \) equals \( 0.184 \).

**Tip**: When working with probability trees, ensure you accurately trace all paths that lead to the event of interest and sum the probabilities of those paths to find the total probability.

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Would you like more details or have any other questions? 

Here are some related questions:

1. How do you determine the paths in a probability tree?
2. What is the significance of conditional probabilities in a probability tree?
3. How would you compute the probability of \(P(\text{not } R)\)?
4. Can you explain the law of total probability with an example?
5. How does Bayes' theorem relate to probability trees?

**Tip**: When constructing probability trees, ensure the branches are mutually exclusive and collectively exhaustive to cover all possible outcomes.

Learn how to compute probabilities using a probability tree with a detailed example. This problem covers finding joint probabilities and conditional probabilities within a probability tree structure. Suitable for Grades 10-12.

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