. Let P(A) = 0.65, P(B) = 0.30, and P(AIB) = 0.45. a. Calculate P(A n B). b. Calculate P(A u B). c. Calculate P(BIA).

1. Break Down the Problem

We need to calculate three probabilities based on the given information:

• $P(A \cap B)$ for part (a)
• $P(A \cup B)$ for part (b)
• $P(B | A)$ for part (c)

2. Relevant Concepts

• Joint Probability: The formula for $P(A \cap B)$ can be given by: $P(A \cap B) = P(B) \cdot P(A | B)$

• Union Probability: The formula for $P(A \cup B)$ is: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Conditional Probability: The formula for $P(B | A)$ is: $P(B | A) = \frac{P(A \cap B)}{P(A)}$

3. Analysis and Detail

a. Calculate $P(A \cap B)$

Using the formula: $P(A \cap B) = P(A | B) \cdot P(B)$ In this case, we substitute: $P(A \cap B) = 0.45 \cdot 0.30 = 0.135$

b. Calculate $P(A \cup B)$

Now, substitute into the union formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ $P(A \cup B) = 0.65 + 0.30 - 0.135 = 0.815$

c. Calculate $P(B | A)$

Using the conditional probability formula: $P(B | A) = \frac{P(A \cap B)}{P(A)}$ Substituting in our values: $P(B | A) = \frac{0.135}{0.65} \approx 0.2077$

4. Verify and Summarize

All calculations have been carried out according to the relevant formulas and checked for accuracy.

Final Answer

a. $P(A \cap B) = 0.135$
b. $P(A \cup B) = 0.815$
c. $P(B | A) \approx 0.2077$