Gaic’ A modified roulette wheel has 40 slots. One slot is 0, another is 00, and the others are numbered through 38, 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 19/40. b. What are the actual odds against winning? The actual odds against winning are 21/19. and then click Check Answer: Enter your answer in the edit fields Clear All parts Check Answer
The Correct Answer and Explanation is:
Problem Setup:
A modified roulette wheel has 40 slots:
- 1 slot is 0
- 1 slot is 00
- 38 slots are numbered 1 through 38.
In roulette, 0 and 00 are not considered odd or even. Only the numbers 1 through 38 are classified as odd or even.
Part (a): Probability of Winning
Odd numbers between 1 and 38 are: 1, 3, 5, …, 37.
This sequence contains 19 odd numbers.
Total number of slots = 40.
Thus, the probability of winning is: P(win)=Number of odd numbersTotal number of slots=1940.P(\text{win}) = \frac{\text{Number of odd numbers}}{\text{Total number of slots}} = \frac{19}{40}.P(win)=Total number of slotsNumber of odd numbers=4019.
Answer for (a):
The probability of winning is 19/40.
Part (b): Actual Odds Against Winning
“Odds against” is calculated as: Odds against=Number of losing outcomesNumber of winning outcomes.\text{Odds against} = \frac{\text{Number of losing outcomes}}{\text{Number of winning outcomes}}.Odds against=Number of winning outcomesNumber of losing outcomes.
Total outcomes = 40
Winning outcomes = 19
Losing outcomes = 40 – 19 = 21
Thus, Odds against=2119.\text{Odds against} = \frac{21}{19}.Odds against=1921.
Answer for (b):
The actual odds against winning are 21/19.
Explanation:
In this modified roulette problem, the wheel consists of 40 distinct slots. Two of these slots, labeled 0 and 00, are neither odd nor even and therefore do not contribute to the count of odd numbers. The remaining 38 slots are numbered sequentially from 1 to 38. Among these, every second number is odd, starting from 1 and increasing by increments of 2. This creates a total of 19 odd numbers.
The probability of winning is determined by comparing the number of favorable outcomes to the total number of possible outcomes. Here, the favorable outcomes are the odd numbers, totaling 19. Since there are 40 total slots on the wheel, the probability of winning becomes 19 divided by 40.
For calculating the actual odds against winning, it is essential to compare the number of losing outcomes to the number of winning outcomes. Losing outcomes include all slots that are not odd. Out of 40 slots, with 19 being odd, the remaining 21 slots represent losing outcomes. These consist of the 19 even numbers between 1 and 38, along with the two slots labeled 0 and 00. Therefore, the odds against winning are 21 to 19.
This distinction between probability and odds is critical. Probability represents the ratio of favorable outcomes to total outcomes, while odds against winning express the ratio of unfavorable outcomes to favorable outcomes. Understanding both measures provides a complete picture of the risks involved in this betting scenario.
