Find P(B) given P(A), P(A∪B)

Short answer

For independent events P(B)=0.5. For mutually exclusive events P(B)=0.3.

Let P(A) = 0.4 and P(A∪B) = 0.7. What is P(B) if: i) A and B are independent? ii) A and B are mutually exclusive?

Given: P(A) = 0.4 and P(A ∪ B) = 0.7. By the addition law, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

For independent events, P(A ∩ B) = P(A) P(B).

0.7 = 0.4 + P(B) − 0.4 P(B)
0.7 − 0.4 = P(B) (1 − 0.4)
0.3 = 0.6 P(B)
P(B) = 0.5

Check: P(A ∩ B) = 0.4 × 0.5 = 0.2, and P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7. ✓

For mutually exclusive events, P(A ∩ B) = 0.

0.7 = 0.4 + P(B)
P(B) = 0.3

Answer: (i) P(B) = 0.5, (ii) P(B) = 0.3.

✓ Answer checked by Claude Sonnet 5.5

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