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.