Expected number of front-facing coins which have each different front-facing probability.

Question

When the number of coins is $n$ and each coin $j$ has each difference front-facing probability $P_j$, how can I find the expected number of front-facing coins?

For example, there is a case that when $n$ is $2$, $P_1$ is $1/3$ and $P_2$ is $1/9$.

Answer 1

Call $X$ the amount of front-facing coins. Then

$$E[X] = \sum_{i=1}^n i P(X = i)$$

Now, you just have to calculate $P(X = i)$. For example,

$$P(X = 1) = P_1 (1-P_2)\dots(1-P_n) + (1-P_1)P_2(1-P_3)\dots (1-P_n) + \dots + (1-P_1)\dots (1-P_{n-1})P_n$$

There might be some closed form of the final expression, but I don't have the time now to look for it.