I'm doing a project on gambling and I've got a set of returns for various bets. I have 219 data points and I'm trying to determine a formula for the probability that returns are positive after $n$ bets.
The mean return is $7.4$ dollars, with a standard deviation of $224.4$ and a variance of $50,358$. My plan originally was just to use the $z$-scores of the distribution and sum them some how but then I realized that I didn't really know what I was doing.
Here is my data set:
-100
-100
-100
-100
-100
425
-100
-100
389
-100
-100
725
-100
-100
470
-100
-100
-100
-100
420
-100
-100
-100
354
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
444
-100
-100
-100
-100
-100
475
-100
-100
500
-100
400
-100
-100
450
-100
-100
460
-100
-100
-100
-100
-100
406
-100
-100
-100
495
390
380
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
435
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
365
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
765
-100
385
-100
415
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
405
-100
540
-100
-100
-100
-100
415
-100
-100
-100
-100
450
-100
-100
-100
-100
-100
-100
-100
373
-100
472
-100
-100
-100
385
-100
700
-100
-100
-100
-100
-100
501
-100
400
-100
-100
-100
590
-100
-100
-100
-100
-100
-100
-100
-100
-100
-100
559
-100
-100
-100
472
-100
444
568
-100
-100
-100
375
-100
500
-100
-100
-100
451
-100
-100
371
-100
-100
-100
-100
-100
-100
-100
-100
361
450
-100
-100
-100
-100
-100
-100
-100
-100
-100
Thanks for any help!
If we assume normality then let$$X_i\sim N(7.4,50358)$$
where the $X_i$'s are independently and identically distributed.
Define
$$Z=X_1+X_2+...+X_n$$
Then
$$\begin{align*} E(Z) &=E(X_1+...+X_n)\\\\ &=E(X_1)+...+E(X_n)\\\\ &=n\cdot E(X_1)\\\\ &=n\cdot7.4 \end{align*}$$
Similarly
$$\begin{align*} Var(Z) &=Var(X_1+...+X_n)\\\\ &=Var(X_1)+...+Var(X_n)\\\\ &=n\cdot Var(X_1)\\\\ &=n\cdot 50358 \end{align*}$$
Then $$Z\sim N(n\cdot7.4, n\cdot50358)$$
and so finally
$$\begin{align*} P(Z\gt0) &=1-P(Z\lt0)\\\\ &=1-\Phi\left(\frac{0-n\cdot7.4}{\sqrt{n\cdot50358}}\right) \end{align*}$$
After closer examination, your data is extremely non-normal so you might want to check if it follows another distribution better. I will let someone more qualified bud in. In the meantime, I suggest you un-accept my answer to increase viewership.