Finds needed for three of four stamp kinds

Question

A collector wants stamps of 4 types, and each stamp she finds is equally likely to be any type, independently. What is the expected number of stamps she must find before she has 3 distinct types?

Short answer

$13/3$ . The first stamp is new; the next new type takes $\tfrac43$ stamps on average (success $\tfrac34$ ), and the third new type takes $2$ (success $\tfrac12$ ). Total $1+\tfrac43+2=\tfrac{13}{3}$ .

Adapted from OpenQuant; 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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