Question

When a roulette wheel is spun once, there are 38 possible outcomes: 18 red, 18 black. and 2 green (if the outcome is green the house wins all bets). If a wheel is spun twice, all \(38 \cdot 38\) outcomes are equally likely. If you are told that in two spins at least one resulted in a green outcome, what is the probability that both outcomes were green?

Short answer

The probability is \(\frac{1}{37}\).

Step-by-step solution

Understanding the Problem

We are told that on a roulette wheel, there are 38 possible outcomes: 18 red, 18 black, and 2 green. If the wheel is spun twice, there are 38 x 38 possible combinations of outcomes. The problem is to find the probability that both outcomes were green given that at least one was green.

Calculate Total Outcomes with at Least One Green

First, calculate the total number of two-spin outcomes where at least one is green. This can be found by subtracting the outcomes with no green from the total outcomes. The total possible outcomes for two spins is 38 * 38 = 1444. The number of outcomes with no green is (36 outcomes/first spin) x (36 outcomes/second spin) = 1296. Thus, the number of outcomes with at least one green is 1444 - 1296 = 148.

Calculate Outcomes with Both Spins Green

Next, calculate the number of two-spin outcomes where both spins are green. Since we have 2 green outcomes each spin, the total outcomes for both spins green is 2 x 2 = 4.

Calculate the Desired Probability

With the number of outcomes calculated, we can now find the probability that both outcomes are green given that at least one is green. Use the conditional probability formula:\[P(\text{both spins green} | \text{at least one green}) = \frac{\text{Number of outcomes where both spins are green}}{\text{Number of outcomes with at least one green}}\]This gives:\[P = \frac{4}{148} = \frac{1}{37}\]

Conclusion

Therefore, the probability that both spins result in green given that at least one spin was green is \(\frac{1}{37}\).

Key concepts

Roulette Probabilities

When you hear the sounds of a roulette wheel spinning, you might wonder about the likelihood of different outcomes. In the game of roulette, there are 38 possible outcomes each time the wheel is spun. These outcomes include 18 red numbers, 18 black numbers, and 2 green numbers. The green numbers are usually marked as 0 and 00.
In a scenario where the roulette wheel is spun twice, you are looking at a much larger set of possible outcomes—exactly 1444. This is calculated by multiplying the 38 possible outcomes from the first spin by the 38 possible outcomes from the second spin.

For calculating the probability of rolling certain outcomes, it's often interesting to determine the likelihood when specific conditions are met. For instance, if you know that at least one green has appeared in the two spins, you may be curious about the probability of both being green. This involves a concept known as conditional probability and makes the world of roulette probabilities both intriguing and strategic.

Probability Theory

Probability theory is the mathematical framework to quantify uncertainty and make informed predictions about future events. It's a branch of mathematics that deals with the study of random events and the likelihood of different outcomes.
At its core, probability theory assigns a numerical value between 0 and 1 to an event, where 0 indicates impossibility and 1 indicates certainty. For example, the probability of getting a green outcome in one spin of the roulette wheel is \( \frac{2}{38}\). For two independent events, like two spins of a roulette wheel, probability theory often involves multiplication or addition rules depending on whether the events are independent or dependent, mutually exclusive, or overlapping.

Conditional probability, a vital part of probability theory, is particularly useful. It deals with the probability of one event occurring given another event has already occurred. The formula for conditional probability is \[P(A | B) = \frac{P(A \cap B)}{P(B)}\]. This allows us to refine probabilities based on additional information, making predictions and decisions more accurate.

Discrete Mathematics

Discrete mathematics provides the tools to deal with distinct and separate values, such as whole numbers or other quantized items. It is the foundation upon which probability theory is built and helps us calculate probabilities in games like roulette.
One key aspect of discrete mathematics in probability is counting the number of elements in a set, which is essential for understanding probabilities. For instance, in the case of the roulette wheel, knowing there are 38 possibilities helps determine the probability of different outcomes.

Another important concept from discrete mathematics is the use of combinatorics to determine the number of possible configurations or outcomes, allowing us to handle complex probability calculations. Without these tools, calculating probabilities in scenarios with multiple possible events happening in sequence, like our spinning roulette wheel, would be daunting. As such, discrete mathematics offers essential techniques for breaking down and solving probability problems, making it indispensable for understanding how games of chance like roulette are analyzed.