Each cereal box has one of $n$ uniformly random coupons. How many boxes until you've seen all $n$? The trick is decomposing the total wait into stages indexed by how many distinct coupons you've already collected.
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Worked example
Cereal boxes contain one of $n = 4$ different prize coupons, each equally likely and independent across boxes. Let $T$ be the number of boxes you buy until you've collected all $4$ distinct coupons. Compute $E[T]$.
✓ Worked example · expand
Practice 1 of 3Type a fraction, decimal, or expression.
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Reflection
Why does linearity of expectation let us sum the stage waits even though the stage durations are dependent on the (random) collection history? What changes if coupons aren't equally likely?