This problem has probably been solved somewhere but I could not find it. We have $n$ Bernoulli random trials $X\_i$ with different occurrence probabilities, $\mathrm{Pr}[X\_i=1]=p\_i>p\_{\min}>0$ for $i=1,2,\dots,n$ and some constant $p\_{\min}$ what is the probability of having even number of ones when $n$ is very large? I guess that this probability is equal to $\frac{1}{2}$ but how can I prove this? I know how to prove this for the case when all $p\_i$'s are equal but how can we prove it for different $p\_i$'s? (In case this helps to solve the problem we can assume that $0<p\_i \le 0.5$.)



Equal probability of having even/odd number of ones in many Bernoulli trials with different probabilities?
The probability that an even number of trials succeed is exactly
$$ \frac12\biggl( 1 + \prod\_{i=1}^n (1-2p\_i)\Biggr).$$
This is a standard elementary application of probability generating functions.


