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I would like to calculate the probability of a certain result coming out at least 'x' times in 'n' attempts when the probability of result varies on every attempt; all attempts are independent events.

I found that I can use PMF when I have the same probability for every event, e.g. probability of 3 tails in 5 coin tosses. But there the probability is 50% (If we use a fair coin) for all events. When the probability changes on each attempt I'm lost.

Let's say that the probabilities of outcomes in 5 events are: 30% in the first, 35% in the second, 50% in the third, 42% for the fourth, and 25% in the fifth. What are the probabilities of it happening at least 1 in the 5, what are the probability of 2 of the 5, 3/5, 4/5 and 5/5?

I don't need the probability of the hypothetical case I'm using here as example just the formula for calculate this kind of probability or the name of this kind of thing so I can search it.

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  • $\begingroup$ Will a recursive formula do? I think a solution in closed form will be exceedingly painful. $\endgroup$ Mar 15 '18 at 15:10
  • $\begingroup$ @StephanKolassa Yes, any approach that yields a correct result is welcomed. I'm coding a script to calculate this kind of probability so I will not be making the calculations by hand, and I'm not calculating big enough sets for performance to be an issue. $\endgroup$
    – Jorge Riv
    Mar 15 '18 at 15:30
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You are looking for the Poisson-Binomial distribution, which precisely models the sum of independent Bernoulli trials that are not necessarily identically distributed.

Calculations tend to become quite cumbersome with this distribution, but you can rely on available algorithms. For example, you can look at R package poibin, which includes a command for the cdf (since you need the probability of at least $x$ successes out of $n$).

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