Conditional Probability

Question

A single card is drawn at random from a standard well-shuffled deck and is observed to be a face card; what is the probability that this drawn card is a king?

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Conditional probability P(A | B) = P(A ∩ B)/P(B) restricts attention to the reduced sample space in which the conditioning event B has already occurred. Here the conditioning event is that the card is a face card, of which a standard deck contains exactly 12 (jacks, queens, and kings across four suits), forming the new effective sample space. This is a textbook JEE Advanced reduced-sample-space situation. Among these 12 face cards, exactly 4 are kings, and since every king is itself a face card, the intersection of "king" and "face card" is just the 4 kings. Therefore the conditional probability equals 4/12 = 1/3. Option 1/4 wrongly conditions on suit rather than on the face-card set. Option 1/13 ignores the conditioning entirely and uses the unconditional chance of drawing a king. Option 4/52 is the raw unconditional probability written before reduction. The governing rule is the restriction of the sample space to the conditioning event, the essence of conditional probability. Plausibility check: 1/3 lies strictly between 0 and 1 and exceeds the unconditional 1/13, which is expected because conditioning on face cards should raise the chance of a king.

Conditional Probability

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