Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find the probability of winning at roulette. Here's our problem statement: A modified roulette wheel has 36 slots. One slot is 0, another is 00, and the others are numbered 1 through 34, respectively. You are placing a bet that the outcome is an odd number in roulette (0 and 00 are neither odd nor even).
OK, the first part of this problem asks for our probability of winning. And to do that, we're going to calculate probability the way we always calculate probability, which is the part over the whole. Well, what's the part? The part is the number of slots that are actually odd, because that's the outcome that we're looking for — how many odd numbers do we have here?
Well, 0 and 00 are not odd, and so we have to look through 1 through 34. Half of those will be odd, and half will be even, so if I take my calculator here and divide 34 by 2, I get 17. So this is the number of odd slots that we have. So here in my answer field, I will set up a fraction. In the top part of the fraction, I put the part; it's the number of odd slots on the wheel.
And then on the bottom of my fraction, I put the total number of slots; this is the whole part, the whole set. So that's gonna be 36. Well done!
Now the second part asks for the actual odds against winning. OK, here we have the odds against winning, which means we're going to calculate this as a fraction, although here it's listed in ratio form. And this first number, which would be the number on top of our fraction, will be the part that's against winning, the part that corresponds with losing. And then on the bottom of the fraction, which corresponds with the second number, that's going to be the part for the winning.
Well, we know what that is; that's going to be the 17. What's the part for the losing? Well, I'm just going to subtract the 17 from the total 36, and that gives me the part that's losing, because you either lose or you're win. So this first part here is going to be 19, and the second part is going to be 17. Well done!
Now, the third part asks for how much profit we make if the payoff odds are one to one and we bet $10 and win. Well, if we win, we're gonna get the $10 back that we bet plus an additional payoff according to the payoff odds, which here is one to one. That means for every $1 we bet, we get $1 of payoff. So if you bet $10, your payoff is gonna be $10. Well done!
Now the last part of the problem asks us the same question but with the caveat that we've somehow convinced the casino to change the payoff odds so that their the same as the actual odds against winning. Well, the odds against winning are right here — 19 to 17. So what we do is we actually convert this to a fraction. We can actually then multiply that by the amount of our bet, and that gives us the amount of our payoff.
That's essentially what we were doing up here in this third part. We took the bet, which is 10, we multiplied it by 1 over 1 — and 1 over 1 is 1, and so anything times 1 is itself — and so the numbers came out the same. Here we're gonna do the same thing, only we're gonna use a different number than 1. So I've got 19 divided by 17. So now I take that and multiply it by my bet, and there's my payoff, which I round to the nearest cent. Good job!
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