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Chapter 4: Problem 42

Answer the given questions that involve odds.A roulette wheel has 38 slots. One slot is 0 , another is 00 , and the others are numbered 1 through \(36,\) respectively. You place a bet that the outcome is an odd number. a. What is your probability of winning? b. What are the actual odds against winning? 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 \(\$ 18 dollars and win? d. How much profit would you make on the \)\$ 18 $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? (Recommendation: Don't actually try to convince any casino of this; their sense of humor is remarkably absent when it comes to things of this sort.)

Short Answer

Expert verified
a. Probability is \(\frac{9}{19}\). b. Odds against winning are \(10:9\). c. Profit is \$18. d. Profit is \$2.

Step by step solution

01

Title - Count the Total Slots

The roulette wheel has 38 slots in total. This includes the numbers 1 through 36, 0, and 00.
02

Title - Determine Odd Number Slots

Out of the numbers 1 through 36, half of them are odd and half are even. So, there are 18 odd numbers (1, 3, 5, ..., 35).
03

Title - Calculate Probability of Winning

The probability of winning when betting on an odd number is the number of odd slots divided by the total number of slots. Therefore, \[ P(\text{winning}) = \frac{18}{38} = \frac{9}{19} \]
04

Title - Determine the Actual Odds Against Winning

Odds against winning are calculated as the ratio of losing outcomes to winning outcomes. There are 20 non-odd slots (18 even, 0, 00). Thus, the odds against winning are \[ \text{Odds against winning} = \frac{\text{Number of non-odd slots}}{\text{Number of odd slots}} = \frac{20}{18} = \frac{10}{9} \]
05

Title - Calculate Profit with Payoff Odds 1:1

If the payoff odds are 1:1 and you bet \$18 and win, you get your original \$18 back plus an additional \$18. Therefore, your profit is \[ \text{Profit} = 18 \text{ dollars} \]
06

Title - Calculate Profit with Actual Odds

If the payoff odds were the same as the actual odds against winning, for a \$18 bet you would receive \[ \text{Payout} = 18 \times \frac{10}{9} = 20 \text{ dollars} \] Therefore, your profit would be \[ \text{Profit} = 20 - 18 = 2 \text{ dollars} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Roulette Wheel Probability
Roulette is a popular casino game that involves a wheel with 38 slots, including numbers 0, 00, and 1 to 36. When you bet on an odd number, you're looking at only half of these 36 numbered slots, which means there are 18 odd numbers (1, 3, 5, ..., 35). The probability of winning a bet on an odd number is calculated by dividing the number of ways you can win by the total number of possible outcomes.
The probability formula is:
\[ P(\text{winning}) = \frac{18}{38} = \frac{9}{19} \]
This means you have roughly a 47.37% chance of winning a bet on an odd number.
Odds Calculation
Understanding odds is crucial in gambling. Odds against winning provide insight into how likely you are to lose compared to winning. To find the odds against winning, you compare the number of losing outcomes to the number of winning outcomes.
In roulette, there are 20 non-odd slots (18 even numbers, 0, and 00) and 18 odd slots. So, the odds against winning are:
\[ \text{Odds against winning} = \frac{\text{Number of non-odd slots}}{\text{Number of odd slots}} = \frac{20}{18} = \frac{10}{9} \]
This ratio shows that for every 9 times you might win, you would expect to lose 10 times.
Payoff Odds
Payoff odds determine how much you stand to profit if you win a bet. In roulette, casinos typically set payoff odds lower than the actual odds to ensure their profitability.
For a bet on an odd number, the typical payoff odds are 1:1. This means if you bet \(18 and win, you receive your \)18 back plus an additional \(18, resulting in a total payout of \)36.
However, the actual odds against winning are \(\frac{10}{9}\), which means if casinos paid according to these actual odds, your \(18 bet would result in a payout of:
\[ 18 \times \frac{10}{9} = 20 \text{ dollars} \]
Thus, your profit would be \)2 if actual odds were used.
Casinos maintain lower payoff odds to ensure they remain profitable over the long term by paying less than the true risk would warrant.

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