Calculate expected end balance

Question

There is a problem like this,

Suppose you have a biased coin with a probability 0.1 of head. You win 100 dollars every time you get a head and lose 50 dollars if it is tail. You start with $0, it is possible to get negative balance. Now, if you throw the coin 100 times, what is your expected end balance?

I know that E(X=x) = x*P(X=x), but in this case, how do we calculate the expected value?

Answer 1

Hint:

• Use the following formula:

$$E\left[\sum_{i=1}^n X_i\right]=\sum_{i=1}^n E\left[X_i\right]$$

Answer 2

Can you compute the expected value of one flip? The linearity of expectation says the expected value of $100$ flips is just $100$ times the expected value of one. This doesn't get you the probability of any given result, but you weren't asked for that.