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I've been reading about the Feynman's Restaurant Problem and found this solution, which apparently is the "standard" one since it's the one cited on the Feynman Lectures website.

I've noticed something weird in the derivation which in my opinion it's not completely right. In the solution there's stated the following:

Similarly if you select $D$ (unique) integers from the range, the expected value of each one is $(N+1)/2$.

But this is not right, since on day $k$ you're not choosing among $N$ integers, but among $N - k$ integers, since you've already tried $k$ dishes. And more importantly, you already know the rank of the $k$ plates you've tried so far. Therefore, on the day $k$ the expected value is no longer $(N + 1) / 2$, but another value - which I don't know how to compute.

Did I miss something? Or am I right that this is not completely rigorous?

Thank you for your help :)

alexmolas
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    Related: https://math.stackexchange.com/questions/296302/how-does-the-answer-to-feynmans-restaurant-problem-change-if-m-is-not-restric –  May 05 '22 at 13:23
  • The statement of the problem says that, when you try a new dish, "all you learn is whether it is the best (highest rated) dish you have tried so far, or not." That contradicts your statement that "you already know the rank of the $k$ plates you've tried so far." – Andreas Blass May 05 '22 at 13:39

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