1. Distribution of X

A game consists of drawing tickets with numbers on them from a box, independently with replacement. In order to play you have to stake $2 each time you draw a ticket. Your net gain is the number on the ticket you draw. Suppose there are 4 tickets in the box with numbers -2, -1, 0, 3 on them. If, for example the ticket shows$3 then you get your stake back, plus an additional $3. a) Let X stand for your net gain in one game. What is the distribution of X? Find E(X) and Var(X). b) If you play 100 times, what is your chance of winning$25 or more?

I'm having a hard time knowing how to figure out the distribution, and therefore am having a hard time with the rest of the problem. It would be great if someone could help me!

2. Originally Posted by bart203
A game consists of drawing tickets with numbers on them from a box, independently with replacement. In order to play you have to stake $2 each time you draw a ticket. Your net gain is the number on the ticket you draw. Suppose there are 4 tickets in the box with numbers -2, -1, 0, 3 on them. If, for example the ticket shows$3 then you get your stake back, plus an additional $3. a) Let X stand for your net gain in one game. What is the distribution of X? Find E(X) and Var(X). b) If you play 100 times, what is your chance of winning$25 or more?

I'm having a hard time knowing how to figure out the distribution, and therefore am having a hard time with the rest of the problem. It would be great if someone could help me!
Let X be the random variable number on ticket, that is, net gain.

It should be plain that Pr(X = -2) = Pr(X = -1) = Pr(X = 0) = Pr(X = 3) = 1/4.

Now use the usual definitions.

3. Hmm ok so E(x) = 0 and Var(x) = 3.5?

But then how do you solve part b?

4. Originally Posted by bart203
Hmm ok so E(x) = 0 and Var(x) = 3.5?

But then how do you solve part b?
Let Y be the random variable net gain after 100 games.

n = 100 is large, so it's reasonable to say that $Y = X_1 + X_2 + .... + X_{100}$ ~ $\text{Normal}\left( \mu = n E(X), \sigma^2 = n Var(X)\right)$.

Calculate $\Pr(Y \geq 25)$.