Question
Roulette: A Nevada roulette wheel has 38 pockets. Eighteen of them are red, eighteen are black, and two are green. Each time the wheel is spun, a ball lands in one of the pockets, and each pocket is equally likely.
(a) What is the probability that the ball lands in a green pocket? Round your answer to four decimal places.
(a) What is the probability that the ball lands in a green pocket? Round your answer to four decimal places.
Answers
GPT-4o mini
To find the probability that the ball lands in a green pocket on a Nevada roulette wheel, we can use the following formula for probability:
\[
P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
\]
In this case, the number of favorable outcomes (the green pockets) is 2, and the total number of outcomes (the total pockets) is 38.
So the probability \( P(\text{green}) \) can be calculated as follows:
\[
P(\text{green}) = \frac{2}{38} = \frac{1}{19}
\]
Now, calculating \( \frac{1}{19} \):
\[
\frac{1}{19} \approx 0.05263157895
\]
Rounding this value to four decimal places:
\[
P(\text{green}) \approx 0.0526
\]
Thus, the probability that the ball lands in a green pocket is
\[
\boxed{0.0526}
\]
\[
P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
\]
In this case, the number of favorable outcomes (the green pockets) is 2, and the total number of outcomes (the total pockets) is 38.
So the probability \( P(\text{green}) \) can be calculated as follows:
\[
P(\text{green}) = \frac{2}{38} = \frac{1}{19}
\]
Now, calculating \( \frac{1}{19} \):
\[
\frac{1}{19} \approx 0.05263157895
\]
Rounding this value to four decimal places:
\[
P(\text{green}) \approx 0.0526
\]
Thus, the probability that the ball lands in a green pocket is
\[
\boxed{0.0526}
\]
(b) If you bet on green on every spin of the wheel, you will lose more than 90% of the time in the long run. Explain why this is so. Round your answer to two decimal places.
You will lose more than 90% of the time in the long run if you always bet on green because [blank] says that in the long run, the percentage of the time you lose will approach [blank] %
You will lose more than 90% of the time in the long run if you always bet on green because [blank] says that in the long run, the percentage of the time you lose will approach [blank] %
GPT-4o mini
You will lose more than 90% of the time in the long run if you always bet on green because the **law of large numbers** says that in the long run, the percentage of the time you lose will approach **94.74%**.
### Explanation:
Here's the breakdown of the reasoning:
1. **Total Number of Outcomes**: In a Nevada roulette wheel, there are 38 pockets: 18 red, 18 black, and 2 green.
2. **Probability of Winning**: The probability of winning (landing on green) is \( \frac{2}{38} = \frac{1}{19} \approx 0.0526 \), or about **5.26%**.
3. **Probability of Losing**: Consequently, the probability of losing (landing on either red or black) is the complement of the probability of landing on green:
\[
P(\text{lose}) = 1 - P(\text{green}) = 1 - \frac{2}{38} = \frac{36}{38} = \frac{18}{19} \approx 0.9474
\]
This translates to around **94.74%**.
Putting it all together:
You will lose more than 90% of the time in the long run if you always bet on green because the **law of large numbers** says that in the long run, the percentage of the time you lose will approach **94.74%**.
### Explanation:
Here's the breakdown of the reasoning:
1. **Total Number of Outcomes**: In a Nevada roulette wheel, there are 38 pockets: 18 red, 18 black, and 2 green.
2. **Probability of Winning**: The probability of winning (landing on green) is \( \frac{2}{38} = \frac{1}{19} \approx 0.0526 \), or about **5.26%**.
3. **Probability of Losing**: Consequently, the probability of losing (landing on either red or black) is the complement of the probability of landing on green:
\[
P(\text{lose}) = 1 - P(\text{green}) = 1 - \frac{2}{38} = \frac{36}{38} = \frac{18}{19} \approx 0.9474
\]
This translates to around **94.74%**.
Putting it all together:
You will lose more than 90% of the time in the long run if you always bet on green because the **law of large numbers** says that in the long run, the percentage of the time you lose will approach **94.74%**.