# Roulette - expected value

#### federico.zampini

##### New Member
Hello,

I want to calculate the expected pay-out of european roulette spins.
Particularly, I want to know what is the expected pay-out at the 200th spins.

I assume the following:
a. the customer's initial amount is $100; b. the customer places a flat bet of$10 each spin [total bet];
c. the customer has the following strategy (and play this each spins)
$1 on number 5$2 on even
$3 on odd$2 on 1-6
$2 on 13-24 d. the customer can inject further money in case he/she goes bankrupt before getting to the 200th spin. I used excel poptools to run Monte Carlo simulations (5000 simulations) and obtain that the average amount of money that the customer has at the 200th spins is$44.
This is in line with the theoretical expected value
(that is, customer's initial amount + 200 * (expected total paid per spin - total bet) = 46).

Now, I have two questions:
1. what is the theoretical expected pay-out if the customer cannot inject further money, that is the customer cannot continue playing if he/she goes bankrupt? How can I calculate this?

2. Would it make sense to run simulations and calculate the average pay-out at the 200th when the money at the 200th might be zero for lots of simulations?

Any suggestion is welcomed. Thank you,
Fede

#### staassis

##### Member
I'm afraid, your question is not perfectly clear. What prevents you from updating your Monte Carlo code to add this extra constraint?

#### federico.zampini

yes, you are right: nothing prevents me from running the simulation. However, my question is "are the results going to make sense if the monte carlo simulation calcualte an average over a truncated distribution (ie truncated because the customer is likely to have $0 at the 200th spin). Thank, Fede #### Dason ##### Ambassador to the humans It's going to give you an estimate of the average payout after 200 spins with your constraints. If you wanted to answer the question without constraints then it wouldn't make sense to use the modified simulation but if you're simulating exactly how the data is generated and if the parameter you want to estimate truly is estimable then you shouldn't ever have an issue with estimating via simulation. #### federico.zampini ##### New Member thanks dason. I still have a question: What is what is the theoretical expected pay-out if the customer cannot inject further money, that is the customer cannot continue playing if he/she goes bankrupt? How can I calculate this? is there any formula? Thank you, Fede #### federico.zampini ##### New Member I think the expected value after 3 repetition can be calculated as follows: n=3 (number of repetitions) p = 0.48 (probability of winning) (deposit)*(n+1)* (p)^n+1 + ((deposit)*(n-1)* (p)^n-1*(1-p))*2 Does anyone know a synthetic formula for this? I mean something that I can use to calculate the expected value at 5 or 10 or 50 repetitions. Thank you, Fede #### hlsmith ##### Omega Contributor This is probably trivial, but say they go into the 23rd round with 8 dollars, how do they allocate it. Using the order above, so they won't make the 13-24 #s bet? #### federico.zampini ##### New Member Let's say they play only on black or red. Initial cash amount is$100.
They place a flat bet of \$10.

If they win they double the amount. If they lose, they have the cash amount - 10.
How can I calculate the expected value at the 50th spin? Is there a formula for this?

Thanks,
Fede