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.

Question

(a) What is the probability that the ball lands in a green pocket? Round your answer to four decimal places.

Answer 1

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

\[ \boxed{0.0526} \]

Answer 2

(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] %

Answer 3

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%.