Question

a modified roulette wheel has 32 slots. one slot is 0, another is 00, and the others are numbered 1 through 30, respectively. you are placing a bet that the outcome is an odd number (in roulette, 0 and 00 are neither odd nor even.)
a. what is your probability of winning?
the probability of winning is \\(\frac{15}{32}\\)
(type an integer or a simplified fraction.)
b. what are the actual odds against winning?
the actual odds against winning are 17 : 15
c. when you bet that the outcome is an odd number, the payoff odds are 1:1. how much profit do you make if you bet $13 and win?
if you win, the payoff is $13
d. how much profit should you make on the $13 bet if you could somehow convince the casino to change its payoff odds so that they are the same as the actual odds against winning?
$\square$ (rounded to the nearest cent as needed.)

Explanation:

Step1: Understand Actual Odds and Payoff

Actual odds against winning are \(17:15\), meaning for every 15 wins, there are 17 losses. Payoff odds equal to actual odds mean for a \$15 bet, you win \$17 (profit is \$17, since you get your bet back plus profit? Wait, no: Payoff odds \(a:b\) mean if you bet \$b and win, you get \$a profit (plus your bet back). Wait, let's clarify.

Wait, actual odds against winning are \(17:15\), so the probability of losing is \(\frac{17}{32}\), winning is \(\frac{15}{32}\). Payoff odds equal to actual odds against winning would be odds in favor of winning? Wait no, odds against winning are \(P(\text{lose}):P(\text{win}) = 17:15\). So if payoff odds are same as actual odds against winning (i.e., odds against winning are \(17:15\), so payoff odds for winning would be \(15:17\)? Wait no, maybe I got it wrong. Wait, odds against an event are \(\frac{P(\text{not } E)}{P(E)}\), so here \(P(E) = \frac{15}{32}\), \(P(\text{not } E) = \frac{17}{32}\), so odds against \(E\) are \(\frac{17}{32} : \frac{15}{32} = 17:15\). So if payoff odds are equal to actual odds against winning, that would mean for a bet of \$15, you win \$17 (because odds against are 17:15, so if you bet on the event, the payoff odds should be based on odds in favor? Wait, no. Wait, when payoff odds are \(a:b\), it means if you bet \$b and win, you receive \$a (profit) plus your bet back. So if the actual odds against winning are \(17:15\), then the odds in favor of winning are \(15:17\). Wait, maybe the problem says "payoff odds so that they are the same as the actual odds against winning". Wait, actual odds against winning are \(17:15\), so payoff odds (for winning) would be \(17:15\)? Wait, no, let's re-express.

Wait, the bet is \$13. Let's recall: If the payoff odds are \(a:b\), then profit = (bet amount) * (\(\frac{a}{b}\)). Wait, let's take the standard definition: Odds against an event \(E\) are \(O(\text{against } E) = \frac{P(\text{not } E)}{P(E)} = \frac{17}{15}\) (since \(P(\text{not } E) = 17/32\), \(P(E) = 15/32\), so ratio is 17/15). So if payoff odds are equal to actual odds against winning, that means for a bet of \$b, the profit is \$a where \(a/b = 17/15\) (since odds against are 17:15, so payoff odds for the event would be such that if you bet \$15, you win \$17 profit). Wait, let's check with the bet amount of \$13.

So we have bet = \$13. We need to find profit. Let's set up the ratio: If odds against are 17:15, then the payoff for a bet of \$15 is \$17 profit (because you bet \$15, win, and get \$15 + \$17 = \$32 back, which is the total outcome? Wait, no, the wheel has 32 slots, so maybe the expected value, but no, payoff odds are based on the odds. Wait, let's use the formula: Profit = (bet amount) * (odds in favor numerator / odds in favor denominator). Wait, no, odds against are 17:15, so odds in favor are 15:17. Wait, I'm confused. Wait, let's read the problem again: "How much profit should you make on the \$13 bet if you could somehow convince the casino to change its payoff odds so that they are the same as the actual odds against winning?"

Wait, actual odds against winning are 17:15. So payoff odds (for winning) would be such that if you bet \$15, you win \$17 (profit), because the odds against are 17:15, meaning for every 15 wins, there are 17 losses. So the casino would pay out \$17 for a \$15 bet to make it fair? Wait, no, payoff odds equal to actual odds against winning: so the payoff odds (the amount you win per bet) should be based on the odds against. So if you bet \$15, and win, you get \$17 profit (so total payout is \$15 + \$17 = \$32). So to find the profit for a \$13 bet, we can set up a proportion.

Let \(x\) be the profit for a \$13 bet. Then, since the ratio of profit to bet should be equal to the ratio of the odds against (17:15), so \(\frac{x}{13} = \frac{17}{15}\). Solving for \(x\): \(x = 13 * \frac{17}{15}\).

Step2: Calculate the Profit

Calculate \(13 * \frac{17}{15}\). Let's compute that: \(13 * 17 = 221\), so \(221 / 15 ≈ 14.7333...\). Rounded to the nearest cent, that's \$14.73. Wait, but let's check again. Wait, maybe I mixed up odds in favor and against. Wait, odds against winning are \(17:15\), so the probability of winning is \(15/32\), losing is \(17/32\). If payoff odds are equal to actual odds against winning, that means the payoff for winning is based on the odds against. So when you bet \$15 (the amount corresponding to the winning probability's denominator), you win \$17 (the amount corresponding to the losing probability's numerator). So the profit is (bet amount) * (odds against numerator / odds against denominator). So for a \$13 bet, profit = \(13 * (17/15)\). Let's compute that: \(13 * 17 = 221\), \(221 ÷ 15 ≈ 14.73\).

Answer:

\boxed{14.73}