You are playing pingpong games with your two friends Jack an

You are playing ping-pong games with your two friends, Jack and Jill. Jill is better than both you and Jack, although there is a non-zero probability that any one person can beat any other person. Jill offers to buy you a muffin if you can win two games in a row out of three games by playing her and Jack alternately. (i.e. You either play Jill-Jack-Jill or Jack-Jill-Jack.) To maximize your chance of winning two games in a row, who should you play first? You must support your decision with mathematics.

Solution

Let the probability of winning against Jill be p1 and against jack be p2.

Given, p1<p2

Let us consider the case of playing Jill-Jack-Jill :

P(Winning 2 out of 3 in a row) = p1*p2*(1-p1) +(1-p1)*p2*p1 + p1*p2*p1 = p1*p2*[1-p1+1-p1+p1] = p1*p2*(2-p1)

Let us consider the case of playing Jack-Jill-Jack :

P(Winning 2 out of 3 in a row) = p2*p1*(1-p2) +(1-p2)*p1*p2 + p2*p1*p2 = p1*p2*[1-p2+1-p2+p2] = p1*p2*(2-p2)

As p1<p2, (2-p1) > (2-p2)

So it is better to play in the order Jill-Jack-Jill