[seqfan] a new sequence idea

[Yifan Xie]

Here is Problem 6 (generalized version) in 2023 AIME I:

There is a random sequence of binary digits containing n 0's and m 1's. Then a person uses the best strategy to guess the digits one-by-one and every digit will be revealed after the person guesses it. Find the mathematical expectation of the number of digits the person guesses correctly.
Suppose that at a certain time the remaining sequence unrevealed consists of j 0's and k 1's, than the person will guess 0 if j>=k, or 1 if j<=k so the possibility that the person guesses correctly is max{j, k}/j+k. Next we should calculate the possibility that the remainder contains such digits, which can be described as C(n-j, n+m-j-k)*C(j, j+k)/C(n, n+m).
However things are actually very complicated and I just gave up trying to develop a simple formula (it is possible for me to write a c++ program).
Another problem is that although the sequence terms are all fractions, T(n, m)'s denominator is proved to be a divisor of C(n, n+m). Would it be better to represent the numerator of the sequence (when the denominator is C(n, n+m)), or to represent the ordinary numerator and denominator in two separate sequences?
Please help.


Best regards,
Yifan Xie (xieyifan4013 at 163.com)


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