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I don’t understand. How does this increase the odds for any one prisoner to better than 50/50? I assume the answer requires later prisoners to know something based on success so far - how does this help with that?
The numbers in the drawers/envelopes are a permutation. If everyone starts with the envelope containing their number and keeps opening the envelope of the card in the previous one, they will all eventually get back to the same number via some sequence.
The sequence (or cycle) each prisoner lands may have length from 1 to 100, and there may be anything from 1 to 100 such sequences. But every prisoner is using the same set of cycles. This strategy means that the fates of the prisoners are no longer independent.
The probability that there's a cycle (there could only ever be at most one) longer than fifty turns out to be about 70%, and that corresponds to a prisoner needing to open more than fifty envelopes to get back to their number.
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