https://file.aiquickdraw.com/mathbot/file/172816208024908vulj94.jpg

Let's go through the questions in the image you uploaded:

### a. What is your probability of winning?
You are betting that the outcome will be an even number. The roulette wheel has 40 slots in total, with numbers 1 through 38 plus two additional slots, 0 and 00. Since 0 and 00 are neither odd nor even, the even numbers are 2, 4, 6, ..., 38.

- Half of the numbers from 1 to 38 are even, so there are $ \frac{38}{2} = 19 $ even numbers.
- Thus, the probability of winning is $ \frac{19}{40} $.

### b. What are the actual odds against winning?
The odds against winning are calculated as:
\[
\text{Odds against winning} = \frac{\text{Number of losing outcomes}}{\text{Number of winning outcomes}}
\]
- There are 19 even numbers (winning outcomes).
- There are 21 losing outcomes (including 19 odd numbers, plus the 0 and 00).

Thus, the odds against winning are $ \frac{21}{19} $, or 21:19.

### c. How much profit do you make if you bet $18 and win, with 1:1 odds?
In this case, the payout is 1:1, meaning if you bet $18 and win, you receive your $18 bet back plus $18 in profit. So, your profit is **$18**.

### d. How much profit should you make on an $18 bet if the payoff odds match the actual odds (21:19)?
If the payoff odds were adjusted to match the actual odds (21:19), the profit would be based on this ratio. The formula for the profit is:

\[
\text{Profit} = \frac{21}{19} \times \text{Bet Amount} - \text{Bet Amount}
\]

Substitute in the $18 bet:

\[
\text{Profit} = \frac{21}{19} \times 18 - 18 = \frac{378}{19} - 18 = 19.89 - 18 = 1.89
\]

So, the profit should be **$1.89**.

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Would you like more details on any part of this? 

Here are 5 related questions for further practice:
1. What would the odds against winning be if the wheel had 42 slots?
2. How would the profit change if the bet amount was $20 instead of $18?
3. How do the odds affect the expected value of the bet?
4. What is the probability of losing a bet on an odd number in this roulette setup?
5. If the odds were 2:1 for an even number, how much would you win on an $18 bet?

**Tip:** When calculating odds, always distinguish between payoff odds (how much you win) and probability odds (chances of winning or losing).

A modified roulette wheel has 40 slots. One slot is 0, another is 00, and the others are numbered 1 through 38. You are placing a bet that the outcome is an even number. (In roulette, 0 and 00 are neither odd nor even.)

Explore how to calculate the probability of winning, actual odds against winning, and payoff amounts in a modified roulette game with 40 slots. Learn about even and odd numbers, and how payoff odds relate to betting strategies. Suitable for high school students studying probability and gambling odds.

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