Given that A, B and C are events such that P(A) = P(B) = P(C) = 1/5, P(A⋂B) = P(B⋂C) = 0 and P(A⋂C) = 1/10. The probability that at least one of the events A, B or C occurs is

Engineering

Mathematics

Addition Theorem on Probability

Sets

Basic Definitions Related to Probability

Question

Given that A, B and C are events such that P(A) = P(B) = P(C) = 1/5, P(A⋂B) = P(B⋂C) = 0 and P(A⋂C) = 1/10. The probability that at least one of the events A, B or C occurs is

1/2

3/5

2/3

2/5

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We have A⋂B⋂C⋂A⋂B
⇒ P(A⋂B⋂C) ≤ P(A⋂B) = 0
Since, P(A⋂B⋂C) ≥ 0, we get P(A⋂B⋂C) = 0.
Now, P (at least one of A, B, C)
= P(A⋂B⋂C) = P(A) + P(B) + P(C) – P(B⋂C) – P(C⋂A) – P(A⋂B) + P(A⋂B⋂C)
$\frac{1}{5} + \frac{1}{5} + \frac{1}{5} - 0 - \frac{1}{10} - 0 + 0 = \frac{1}{2}$ .