Question

Let A and B be events with P(A)=0.7, P(B)= 0.3 and P(A and B) = 0.3.

(a) Are A and B independent? Explain.

(b) Compute P(A or B).

(c) Are A and B mutually exclusive? Explain.

Answer 1

To answer your questions about events A and B, we start by recalling some definitions and using the provided probabilities.

Given:
- \( P(A) = 0.7 \)
- \( P(B) = 0.3 \)
- \( P(A \cap B) = P(A \text{ and } B) = 0.3 \)

### (a) Are A and B independent? Explain.

Two events A and B are independent if the probability of them occurring together is equal to the product of their probabilities:
\[
P(A \cap B) = P(A) \cdot P(B)
\]

Calculating the right-hand side:
\[
P(A) \cdot P(B) = 0.7 \cdot 0.3 = 0.21
\]

Now, we compare this with \( P(A \cap B) \):
\[
P(A \cap B) = 0.3
\]

Since \( 0.3 \neq 0.21 \), the events A and B are **not independent**.

### (b) Compute \( P(A \text{ or } B) \).

The probability of A or B can be computed using the inclusion-exclusion principle:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Substituting in the values:
\[
P(A \cup B) = 0.7 + 0.3 - 0.3 = 0.7
\]

Thus, the probability of either A or B occurring is:
\[
P(A \text{ or } B) = 0.7
\]

### (c) Are A and B mutually exclusive? Explain.

Two events A and B are mutually exclusive if they cannot occur at the same time, which means:
\[
P(A \cap B) = 0
\]

Given that \( P(A \cap B) = 0.3 \), which is greater than 0, we conclude that events A and B are **not mutually exclusive**.

In summary:
(a) A and B are not independent.
(b) \( P(A \text{ or } B) = 0.7 \).
(c) A and B are not mutually exclusive.