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Official PYQ · 2011Q3Conditional Probability and Bayes TheoremMCQ1 mark

GATE CS 2011 Q3 — Conditional Probability and Bayes Theorem

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If two fair coins are flipped and at least one of the outcomes is known to be a head, what is the probability that both outcomes are heads?

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Answer: A

Let the sample space for flipping two fair coins be:

S={HH,HT,TH,TT}S = \{HH, HT, TH, TT\}

We define two events:

  • Event BB: At least one outcome is a head.
    B={HH,HT,TH}B = \{HH, HT, TH\}
  • Event AA: Both outcomes are heads.
    A={HH}A = \{HH\}

We need to find the conditional probability that both outcomes are heads given that at least one is a head, which is P(A∣B)P(A \mid B):

P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Since A∩B={HH}A \cap B = \{HH\}, we have:

  • P(A∩B)=14P(A \cap B) = \frac{1}{4}
  • P(B)=34P(B) = \frac{3}{4}

Substituting these values into the formula:

P(A∣B)=1434=13P(A \mid B) = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}

Thus, the correct option is A.

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