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fair coin is to be flipped 100 times

Source: Polish MO Finals 1980 p3

August 24, 2024
combinatoricsprobability

Problem Statement

Let kk be an integer in the interval [1,99][1,99]. A fair coin is to be flipped 100100 times. Let εj={1,if the j-th flip is a head2,f the j-th flip is a tail\varepsilon_j =\begin{cases} 1, \text{if the j-th flip is a head} \\ 2, \text{f the j-th flip is a tail}\end{cases} Let MkM_k denote the probability that there exists a number ii such that k+ε1+...+εi=100k+\varepsilon_1 +...+\varepsilon_i = 100. How to choose kk so as to maximize the probability MkM_k?
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