# Thread: Expected amount a person wins/looses after m games at a casino

1. ## Expected amount a person wins/looses after m games at a casino

Hi everyone,

Assume that Jack goes to a casino. He plays n games. Every time he wins a game he gets 100$and every time he looses a game he looses 50$. The probability that he wins i-th game is 1/i. In other words, the probabilities of winning first five games are 1,1/2,1/3,1/4,1/5, respectively.

Can we compute the expected amount of money he has won or lost after he
has played n games? An easier version will be how can we compute the expected amount after he has played 5 games.

Best Regards,
mac

2. ## Re: Expected amount a person wins/looses after m games at a casino

Originally Posted by mac154
Hi everyone,

Assume that Jack goes to a casino. He plays n games. Every time he wins a game he gets 100$and every time he looses a game he looses 50$. The probability that he wins i-th game is 1/i. In other words, the probabilities of winning first five games are 1,1/2,1/3,1/4,1/5, respectively.

Can we compute the expected amount of money he has won or lost after he
has played n games? An easier version will be how can we compute the expected amount after he has played 5 games.

Best Regards,
mac
The expected return on the $i$ th game is

$E(r_i)= -50\left(1-\frac{1}{i}\right)+100\left( \frac{1}{i}\right)$

Therefore the expected total return after $n$ games is:

$E(R_n)=E\left(\sum_{i=1}^n r_i\right)=\sum_{i=1}^nE(r_i)$

CB

Many thanks