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Official PYQ · 1994Q2.8Probability: Random Variables and DistributionsNAT1 mark

GATE CS 1994 Q2.8 — Probability: Random Variables and Distributions

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Let A,B,A, B, and CC be independent events which occur with probabilities 0.8,0.5,0.8, 0.5, and 0.30.3 respectively. The probability of occurrence of at least one of the event is _______
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Answer: 0.93

Given the probabilities of independent events A,B,A, B, and CC:

  • P(A)=0.8  ⟹  P(A′)=1−0.8=0.2P(A) = 0.8 \implies P(A') = 1 - 0.8 = 0.2
  • P(B)=0.5  ⟹  P(B′)=1−0.5=0.5P(B) = 0.5 \implies P(B') = 1 - 0.5 = 0.5
  • P(C)=0.3  ⟹  P(C′)=1−0.3=0.7P(C) = 0.3 \implies P(C') = 1 - 0.3 = 0.7

Since the events are independent, the probability that none of the events occur is the product of their complement probabilities:

P(None)=P(A′)×P(B′)×P(C′)P(\text{None}) = P(A') \times P(B') \times P(C')P(None)=0.2×0.5×0.7=0.07P(\text{None}) = 0.2 \times 0.5 \times 0.7 = 0.07

The probability of occurrence of at least one of the events is:

P(At least one)=1−P(None)=1−0.07=0.93P(\text{At least one}) = 1 - P(\text{None}) = 1 - 0.07 = 0.93

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