This is actually an interesting problem. I was inspired to write a blog post about it covering in detail fair vs unfair coin tosses all the way to the OP's situation of having a different probability for each coin. You need a technique called dynamic programming to solve this problem in polynomial time.
General Problem: Given C, a series of n coins p1 to pn where pi represents the probability of the i-th coin coming up heads, what is the probability of k heads coming up from tossing all the coins?
This means solving the following recurrence relation:
P(*n*,k,*C*,i) = pi x P(*n*-1,k-1,C,*i*+1) + (1-pi) x P(*n*,k,*C*,i+1)
A Java code snippet that does this is:
private static void runDynamic() {
long start = System.nanoTime();
double[] probs = dynamic(0.2, 0.3, 0.4);
long end = System.nanoTime();
int total = 0;
for (int i = 0; i < probs.length; i++) {
System.out.printf("%d : %,.4f%n", i, probs[i]);
}
System.out.printf("%nDynamic ran for %d coinsin %,.3f ms%n%n",
coins.length, (end - start) / 1000000d);
}
private static double[] dynamic(double... coins) {
double[][] table = new double[coins.length + 2][];
for (int i = 0; i < table.length; i++) {
table[i] = new double[coins.length + 1];
}
table[1][coins.length] = 1.0d; // everything else is 0.0
for (int i = 0; i <= coins.length; i++) {
for (int j = coins.length - 1; j >= 0; j--) {
table[i + 1][j] = coins[j] * table[i][j + 1] +
(1 - coins[j]) * table[i + 1][j + 1];
}
}
double[] ret = new double[coins.length + 1];
for (int i = 0; i < ret.length; i++) {
ret[i] = table[i + 1][0];
}
return ret;
}
What this is doing is constructing a table that shows the probability that a sequence of coins from pi to pn contain k heads.
For a deeper introduction to binomial probability and a discussion on how to apply dynamic programming take a look at Coin Tosses, Binomials and Dynamic Programming.