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I am given a sequence of $40$ ones and zeros and I have to test the null hypothesis that ${40 \choose n_1}$ sequences are all equally probable ($n_1$ being the number of ones). To do so, I have to use the number of times that 1 became 0 and vice versa and then apply normal approximation.

What I did so far is define $Z:=1_{X_j \neq X_{j+1}}$ and then the number of 0-1 and 1-0 changes is $I:= \sum_{j=1}^{n-1}Z_j$ (which is binomial); I then calculated its expectation. From here on, I am stuck. What does it mean to use $I$ to show that "${40 \choose n_1}$ sequences are all equally probable"?

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    $\begingroup$ This makes little sense and doesn't appear to be testable from just one sequence. Plotting the random walk given by $I$ can be helpful. See the closely related thread at stats.stackexchange.com/questions/574333 for illustrations and more ideas. $\endgroup$
    – whuber
    May 12 at 19:04

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