Answering BRILLIANT problems

Well, there is a logical answer.

If the answer was 0, then the answer could also be 10, as they have the same probability, and this would not be a correct problem as it has 2 answers.

If the answer was 1, then the answer could also be 9, as they have the same probability, and this would not be a correct problem as it has 2 answers.

And so on...

The only number which does not have another of equal probability is 5, which means the answer is 5, as it is the only answer which would not be ambiguous.

This makes me want to write a program. The logical answer is that more streaks has increasingly high probabilities.

Allowing p(x=5) may have $\binom{10}{5}=252$ possible ways, but this includes both alternating series of (1,0,1,0,1,0,1,0,1,0) and (0,1,0,1,0,1,0,1,0,1) that has 0 streaks.

These series both coming with a probability $\prod_{k=1}^{10} 2^{-k}=2^{-\sum_{k=1}^{10} k}=2^{-55}$ .

We could also see that P(x=k)=P(x=10-k) and p(x=1)<p(x=0) and thus min(p) happens at k=5.

Using excel, I found the percents are as follows

P(x=0) = 28.907%

P(x=1) = 9.808%

P(x=2) = 4.960%

P(x=3) = 3.061%

P(x=4) = 2.253%

P(x=5) = 2.023%

P(x=6) = 2.253%

P(x=7) = 3.061%

P(x=8) = 4.960%

P(x=9) = 9.808%

P(x=10) = 28.907%

I tried 0 and 10, and then I realized it's a trick question and guessed 5. nothing very complicated, just guessing...