Expected cards to complete a four-card set

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A game gives one of four distinct cards per pack, each equally likely and independent. You open packs until you own all four cards. What is the expected number of packs you must open?

Accepted Answer

With j of the four cards held, a new pack is new with probability \frac{4 - j}{4}. The expected packs to advance from stage j satisfies ej = 1 + \frac{j}{4} ej, so ej = \frac{4}{4 - j}. Summing over j = 0,1,2,3: E = \frac44 + \frac43 + \frac42 + \frac41 = 1 + 	frac43 + 2 + 4 = \frac{25}{3} \approx 8.33. Takeaway. Four types cost 4(1 + 	frac12 + 	frac13 + 	frac14) = 	frac{25}{3} packs; the last, rarest card dominates the wait.

Expected cards to complete a four-card set

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