Suppose there are two events, A and B such that
P(AMB) = 0.15,
P(AN B') = 0.2,
P(A'MB) = 0.1, and
P(ANB') = 0.55.
Use the Law of Total Probability to answer the following.
Find P(A).
Find P(B).

We are given the following probabilities:

- \( P(A \cap B) = 0.15 \)
- \( P(A \cap B') = 0.2 \)
- \( P(A' \cap B) = 0.1 \)
- \( P(A' \cap B') = 0.55 \)

### Step 1: Find \( P(A) \)

Using the Law of Total Probability, \( P(A) \) can be written as:

\[
P(A) = P(A \cap B) + P(A \cap B')
\]

Substituting the values:

\[
P(A) = 0.15 + 0.2 = 0.35
\]

### Step 2: Find \( P(B) \)

Similarly, \( P(B) \) can be found using the Law of Total Probability:

\[
P(B) = P(A \cap B) + P(A' \cap B)
\]

Substituting the given values:

\[
P(B) = 0.15 + 0.1 = 0.25
\]

Thus, the probabilities are:
- \( P(A) = 0.35 \)
- \( P(B) = 0.25 \)

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Do you have any questions or need further details?

Here are five related questions for further exploration:
1. How would you calculate \( P(A \cup B) \)?
2. What is the probability \( P(A' \cap B') \), and how is it used in this scenario?
3. If \( A \) and \( B \) were independent events, what would be the relationship between \( P(A) \) and \( P(B) \)?
4. How can you calculate \( P(B') \) from the given information?
5. What is the significance of the Law of Total Probability in probability theory?

**Tip:** When calculating probabilities, always ensure that the total probability sums to 1 as a sanity check.

Suppose there are two events, A and B such that P(A∩B) = 0.15, P(A∩B') = 0.2, P(A'∩B) = 0.1, and P(A∩B') = 0.55. Use the Law of Total Probability to answer the following. Find P(A). Find P(B).

Solve a probability problem involving events A and B using the Law of Total Probability. Given values for P(A∩B), P(A∩B'), P(A'∩B), and P(A∩B'), find P(A) and P(B) step-by-step. Suitable for high school students in Grades 10-12.

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