Practice section 7 of 7

A player wins by balancing a beam with three alternating items: ball–cube–ball or cube–ball–cube, as shown. An unbiased die determines each item: scores 1 or 2 give a ball; otherwise a cube. The player has at most 5 throws. An item of the same type as the last accepted item is rejected.

Let X be the number of throws required to win. Show that $P(X=4)=2/9$. [3]

Find the probability of winning. [5]

The entry fee is a dollars. A win on throw 3 pays back 2 a dollars, on throw 4 pays back a/2 dollars, and on throw 5 pays back a/3 dollars. Find a if the owner expects a profit of 1.20 dollars per game. [4]