# Modelling using the Binomial Distribution Formula

So I'm in my first semester at University studying Actuarial Science and one of my classes is Probability. Needless to say, I have fallen in love with the whole topic and given my passion, I constantly try to solve problems on my own. Except that this time, I really am puzzled.

I first started with a typical American Roulette game at the casino (zero and double zero are possible outcomes). I was interested in finding how many times would the casino need to spin the wheel before being almost sure of turning a profit; assuming there is only 1 bet per spin, the player always bets on red and the bet amount is always 100$.

Well, this problem is not too difficult to solve. Intuitively we know that because the casino has an edge (expected value of 5,26$ for 100$ spin), it will beat the player over the long haul and turn a profit. But knowing the player could get lucky, and delay the inevitable (going bankrupt), how long could the "long haul" be? Would it be 10 spins? 50 spins? 100 spins?

As it turns out, I did the model for this projection and noticed that after 10 spins, the casino will only turn a profit 44.32% of the time. After 100 spins, it becomes 66.57% and after 1000 spins it is 94.89%. Conclusion, our intuition was confirmed. Over time, chances are the casino will crush the player.

I was able to do this using the Binomial Distribution Formula below:

Now, what if instead, my new game at the casino was to roll a fixed die. Possible outcomes would include (1,2,3,4,5,6) and their corresponding probabilities would be (10%,15%,20%,15%,10%,30%). Each number has the following profits for the casino (-200$,-100$,-50$,-100$,-200$,500$).

As you can see, the casino only makes money when a 6 is rolled (30% of the time), and although it doesn't win as often as with the Roulette game, its expected value is even higher (70$ per roll).

How would I find out how many events it would take for the casino to be profitable in this game? Intuitively I'm puzzled because the lower chance of winning tells me it would take much longer than the Roulette game but it's expected value being much higher tells me it would take fewer events.

Can I use a Binomial Distribution once again? For instance, after 5 rolls, 1 possible result could be (5,4,6,3,1) and in that case, the casino would have lost 50$ (-200$,-100$,500$,-50$,-200$). And it's confusing because unlike the Roulette game, the casino has 4 distinct outcomes (-200$,-100$,-50$ & 500$). I don't know how to solve this.

Any help? :)