Question

A new promotion at Insomnia Cookies lets you play the following game: by paying a fixed amount of money, you can get a free cookie every day, selected uniformly at random from chocolate chunk, double chocolate mint, and peanut butter chip, until you get a chocolate chunk (after which the promotion will stop). Suppose you decide to play this game.

(a)Let X be the number of free cookies you obtain with this promotion. Find Ex(X).

(b)Let Y be the number of free cookies you get that are not chocolate chunk. Find Ex(Y).

Answer 1

a. X is the number of free cookies I get.

So X takes the values 1,2,3,4,5,......

Now,

P(X=1)=1/3. I get the chocolate chunk.

P(X=2)=2/3*1/3. I get any one of double chocolate mint or peanut butter chip on first day and chocolate chunk the next day.

Similarly,

P(X=3)=2/3*2/3*1/3

So, E(X)=1*1/3+2*2/3*1/3+3*2/3*2/3*1/3+....

Now Y takes the values 0,1,2,3,...

P(Y=0)=1/3. You get chocolate chunk at first attempt.

P(Y=1)=2/3*1/3. You miss the first time and get chocolate chunk in the second time. and so on.

So E(Y) equals,

E(Y)=1*2/3*1/3+2*2/3*2/3*1/3+...