summation subsets formula

Question

I would like to know the explanation of this formula please, does it depend on Binomial distribution ?

$\sum_{A\subseteq S} P ^ {|A|} (1-P)^{|S|-|A|} =1.$

Answer 1

Not on the binomial distribution but on the binomial theorem. Remember that the binomial theorem says that

$$\sum_{k=0}^n\binom n k x^ky^{n-k} = (x+y)^n.$$

Can you see how to get to your formula from using $x=P$ and $y=1-P$ above?
Hint: For each $k$, how many subsets of $S$ with size $k$ are there?