GATE CS 2011 Q.3 (CS Q.3): Probability and Statistics

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?

$1/3$

$1/4$

$1/2$

$2/3$

Official answer and solution

Solution

Apply conditional probability by identifying the reduced sample space given that at least one outcome is a head.

When two fair coins are flipped, the complete sample space is $S = \{HH, HT, TH, TT\}$ , with each of the 4 elementary outcomes having probability $1/4$ .

The conditioning event $E$ is that at least one coin shows a head: $E = \{HH, HT, TH\}$ , which contains $|E| = 3$ outcomes.

The target event $A$ is that both coins show heads: $A = \{HH\}$ . The intersection is $A \cap E = \{HH\}$ .

P(A \mid E) = \frac{|A \cap E|}{|E|} = \frac{1}{3}

Final answer

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