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2 persons shoot it each other. Person A shoots at Person B, if A misses, B shoots at A and so on.

The Game continues until one of them hits the other one.

Probability that A hits B is P1 ,and probability that B hits A is P2.

I need to find the E[X].

They give some guidance:

Declare another Random variable Y as follows:

Y=0 if person A hits person B on fist shot.

Y=1 if person B hits person A on fist shot.

Y=2 if none of them hits on first shot.

I also have a solution, that I don't understand:

The idea is: E(X)=E[E[X|Y]]=[tex]\sum[/tex]E[X|Y=y]P(y) (this I understand)

What I don't understand is how they've calculated the following:

g(0)=E[X|Y=0]=1

g(0)=E[X|Y=1]=2

g(0)=E[X|Y=2]=2+E[X]