The suit that shows up last

Question

A machine outputs one of 4 equally likely category tokens per press (call them the four suits), independent each time. You press until all four suits have appeared. One suit is hearts.

What is the probability that hearts is the last suit to appear for the first time?

Accepted Answer

Every suit is produced with the same probability on every press, so the four suits are interchangeable. Naming one of them "hearts" changes nothing about the process. By symmetry, each suit is equally likely to be the last new one, so P(	ext{hearts is last}) = \frac{1}{4} = 0.25. Takeaway. With n equally likely types, each is last with probability \frac{1}{n}; for four suits that is \frac{1}{4}.

The suit that shows up last

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