Card after the first ace at position fourteen

Question

A standard 52-card deck is shuffled and dealt face up one card at a time. The first ace turns up as the 14th card. Given this, what is the probability that the 15th card is the king of hearts?

Short answer

$35/1824$ . The 13 cards before the ace are a random set of 13 of the 48 non-aces, so the king of hearts avoids them with probability $\tfrac{35}{48}$ . If it does, it is equally likely to be any of the 38 cards after position 14, giving $\tfrac{35}{48}\cdot\tfrac{1}{38}=\tfrac{35}{1824}$ .

Adapted from OpenQuant; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Derivation — Means and Variances of the Named Distributions
Sample Spaces, Events, and the Probability Axioms

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