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How to calculate probability of the same events happening 3 times in a row, from binomial distribution given n trials?

Laszlo Benedek
Last Modified: 2016-02-20
In a game (XCOM) you have a stated chance of hitting the target. You either hit or miss. Let's say your chance of hitting is 90% every time.

I want to know the chance of 3 misses (10% each) happening 3 times in a row anywhere given n trials (let's say 50, 100, 200 etc)

Ideally I'm looking for a simple solution, and it can be an approximation.

The only way I can think is working out permutations for each possible number of misses (probability of x number of misses calculated from binomial formula) but that would take quite a long time. I'm wondering if there's a more elegant way.

It would be also nice to know the probability of at least 1 such streak, at least 2 etc.
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