Question:
Assume the following probabilities for two events, A and B: P(A) = 0.50,P(B) = 0.50, and P(A ∪ B) = 0.85. Then we can conclude that
Assume the following probabilities for two events, A and B: P(A) = 0.50,P(B) = 0.50, and P(A ∪ B) = 0.85. Then we can conclude that
Updated On: Aug 21, 2025
- A and B are mutually independent
- A and B are equally likely
- A and B are not mutually independent
- A and B are mutually exclusive
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The Correct Option is A
Solution and Explanation
To determine the relationship between events A and B, we need to check if they are mutually independent. Two events A and B are considered independent if:
P(A ∩ B) = P(A) * P(B)
We are provided with the following probabilities:
- P(A) = 0.50
- P(B) = 0.50
- P(A ∪ B) = 0.85
Using the principle of inclusion-exclusion for probabilities, we know:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Substituting the given values:
0.85 = 0.50 + 0.50 - P(A ∩ B)
0.85 = 1.00 - P(A ∩ B)
P(A ∩ B) = 1.00 - 0.85
P(A ∩ B) = 0.15
Next, we calculate P(A) * P(B):
P(A) * P(B) = 0.50 * 0.50 = 0.25
Since P(A ∩ B) = 0.15, which is not equal to P(A) * P(B) = 0.25, events A and B are not mutually independent.
The correct conclusion is: A and B are not mutually independent.
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