Different songs in a short shuffle

Question

A "random" shuffle picks each track independently and uniformly from 10 songs (repeats allowed). It plays 5 tracks. On average, how many distinct songs will you hear across the 5 tracks?

Accepted Answer

By linearity of expectation, add up over the 10 songs the probability that each is played at least once. A specific song is skipped on one track with probability \frac{9}{10}, so it is absent from all 5 tracks with probability \left(\frac{9}{10}ight)^5 = 0.59049. It plays at least once with probability 1 - 0.59049 = 0.40951. With 10 songs, E[	ext{distinct}] = 10\left(1 - \left(	frac{9}{10}ight)^5ight) \approx 10 	imes 0.40951 = 4.095. Roughly 4 different songs from 5 tracks, so repeats are already common. Takeaway. Expected distinct types after m draws is n\left(1 - \left(\frac{n-1}{…

Different songs in a short shuffle

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