Distinct reactions after ten clicks

Question

On a post, 10 people each add one reaction, chosen independently and uniformly from 6 emoji options. On average, how many distinct emojis appear among the 10 reactions?

Accepted Answer

By linearity of expectation, the expected number of distinct emojis is the sum over the 6 emojis of the chance that each appears at least once. A specific emoji is absent from all 10 reactions with probability \left(\frac{5}{6}ight)^{10} \approx 0.1615, so it appears with probability 1 - 0.1615 = 0.8385. With 6 emojis, E[	ext{distinct}] = 6\left(1 - \left(	frac56ight)^{10}ight) \approx 6 	imes 0.8385 = 5.031. Takeaway. Expected distinct types after m draws is n\left(1 - \left(\frac{n-1}{n}ight)^might); here about 5 of the 6 emojis show up.

Distinct reactions after ten clicks

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