Compute the indicated probabilities by referring to the probability tree.

​(A)**** 

​P(Mintersect∩​S)

​(B)**** ​P(R)

0.80.20.30.70.70.3

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Part 1

​(A)**** 

​P(Mintersect∩​S)equals=enter your response here  

​(Type an integer or a​ decimal.)




It seems like the question involves a probability tree, but the description provided doesn't fully explain it. Based on the data you've given, I'll guide you through the process of solving this using a probability tree approach.

### (A) P(M ∩ S)

To compute \( P(M \cap S) \), which is the probability of both events \( M \) and \( S \) occurring, we need to multiply the probabilities along the branches of the tree that lead to \( M \) and \( S \).

Let's assume the structure is as follows:

1. Probability of event \( M \) is 0.80.
2. If \( M \) occurs, the probability of \( S \) occurring after \( M \) is 0.70.
3. If \( M \) does not occur, the probability of \( S \) occurring after \( \text{not M} \) is 0.30.

Thus:
\[
P(M \cap S) = P(M) \times P(S|M)
\]
Substitute the values:
\[
P(M \cap S) = 0.80 \times 0.70 = 0.56
\]

So, \( P(M \cap S) = 0.56 \).

### (B) P(R)

If \( P(R) \) is provided as part of the probability tree (possibly referring to another event after some initial event), you would use a similar method as for \( P(M \cap S) \). However, since no details about \( R \) were given, I cannot calculate it directly. If you provide the probabilities related to \( R \), I can help compute it.

Would you like more details or have any questions?

### Relative Questions:
1. How would the result change if \( P(S|M) \) was different?
2. What does \( P(M^c \cap S) \) represent in this scenario?
3. How do you calculate the total probability of \( S \) occurring?
4. Can you determine \( P(S) \) using the law of total probability?
5. What is the significance of the intersection \( P(M \cap S) \) in probability theory?

### Tip:
When dealing with probability trees, always follow the branches from start to end for the event of interest, multiplying probabilities along the way to determine the overall likelihood of combined events.

Learn how to compute probabilities using a probability tree. This problem involves calculating the intersection probability of events M and S, along with understanding conditional probabilities. Suitable for high school students.

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