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Official PYQ · 1997Q1.1Probability: Random Variables and DistributionsMCQ1 mark

GATE CS 1997 Q1.1 — Probability: Random Variables and Distributions

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The probability that it will rain today is 0.50.5. The probability that it will rain tomorrow is 0.60.6. The probability that it will rain either today or tomorrow is 0.70.7. What is the probability that it will rain today and tomorrow?

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Answer: D

Let AA be the event that it rains today, and BB be the event that it rains tomorrow.

We are given the following probabilities:

  • P(A)=0.5P(A) = 0.5
  • P(B)=0.6P(B) = 0.6
  • P(A∪B)=0.7P(A \cup B) = 0.7

Using the addition theorem of probability:

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Substitute the given values into the formula:

0.7=0.5+0.6−P(A∩B)0.7 = 0.5 + 0.6 - P(A \cap B)0.7=1.1−P(A∩B)0.7 = 1.1 - P(A \cap B)P(A∩B)=1.1−0.7=0.4P(A \cap B) = 1.1 - 0.7 = 0.4

Thus, the probability that it will rain both today and tomorrow is 0.40.4, which corresponds to option D.

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