Math Problem Statement

Compute the indicated probabilities by referring to the probability tree.

​(A)****

​P(Mintersect∩​S)

​(B)**** ​P(R)

0.30.70.40.60.60.4

Question content area bottom

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.)

Solution

To compute the probabilities using the probability tree, let's assume you have a tree with events MM, SS, and RR with the corresponding probabilities given.

Let's break down the problem:

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

You've already calculated P(MS)=0.18P(M \cap S) = 0.18. This is the probability that both events MM and SS happen simultaneously.

Part B: P(R)P(R)

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

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

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

Let's say you have two paths leading to RR as an example:

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

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

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

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

P(R)=0.3×0.6×0.4+0.7×0.4×0.4P(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=0.184P(R) = 0.072 + 0.112 = 0.184

So, P(R)P(R) equals 0.1840.184.

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.


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(not R)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.

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Conditional Probability
Probability Trees

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12