Expected position of the final heart in a shuffled deck

Question

A well-shuffled $52$ -card deck is turned over one card at a time. What is the expected number of cards turned over up to and including the moment the thirteenth (last) heart appears?

Short answer

$\tfrac{689}{14}$ (about 49.2). The 13 hearts cut the 39 other cards into 14 gaps, and by symmetry each gap holds $\tfrac{39}{14}$ cards on average. Only the final gap lies after the last heart, so its expected position is $52-\tfrac{39}{14}=\tfrac{689}{14}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Sample Spaces, Events, and the Probability Axioms

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