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Official PYQ · 2010Q26Probability: Random Variables and DistributionsMCQ1 mark

GATE CS 2010 Q26 — Probability: Random Variables and Distributions

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Consider a company that assembles computers. The probability of a faulty assembly of any computer is pp. The company therefore subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of qq. What is the probability of a computer being declared faulty?

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Answer: A

We can solve this problem by applying the law of total probability. Let us define the events as follows:

  • Let FF be the event that a computer is actually faulty. Thus, P(F)=pP(F) = p and P(Fc)=1−pP(F^c) = 1 - p.
  • Let TT be the event that the testing process declares the computer as faulty.

The testing process gives the correct result with a probability of qq. This means:

  • If the computer is actually faulty (FF), the correct declaration is "faulty". Hence, P(T∣F)=qP(T \mid F) = q.
  • If the computer is not faulty (FcF^c), the correct declaration is "not faulty". Thus, the incorrect declaration "faulty" occurs with probability P(T∣Fc)=1−qP(T \mid F^c) = 1 - q.

Using the law of total probability, the probability that a computer is declared faulty is:

P(T)=P(T∣F)P(F)+P(T∣Fc)P(Fc)P(T) = P(T \mid F)P(F) + P(T \mid F^c)P(F^c)

Substituting the respective probabilities:

P(T)=q⋅p+(1−q)⋅(1−p)=pq+(1−p)(1−q)P(T) = q \cdot p + (1 - q) \cdot (1 - p) = pq + (1 - p)(1 - q)

Therefore, the correct option is A.

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