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Roulette-Betting on Red. A roulette wheel has 38 slots, numbered 0,00 , and 1 to 36 . The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors. a. If you bet on "red," you win if the ball lands in a red slot. What is the probability of winning with a bet on red in a single play of roulette? b. You decide to play roulette four times, each time betting on red. What is the distribution of \(X\), the number of times you win? c. If you bet the same amount on each play and win on exactly two of the four plays, then you will "break even." What is the probability that you will break even? d. If you win on fewer than two of the four plays, then you will lose money. What is the probability that you will lose money?

Step by step solution

01

Calculate Probability of Winning on Red

To find the probability of winning with a bet on red, we first note there are 18 red slots on the roulette wheel out of a total of 38 slots. Therefore, the probability of landing on a red slot (and thus winning when betting on red) is given by \( P( ext{Red}) = \frac{18}{38} \). Simplifying the fraction gives us \( P( ext{Red}) = \frac{9}{19} \approx 0.474 \).

02

Determine Distribution of Wins

The number of times you win, \( X \), when betting four times on red follows a binomial distribution, since each play is independent and has a constant probability of success. The parameters for this distribution are \(n = 4\) and \(p = \frac{9}{19}\). The probability mass function is given by \( P(X = k) = \binom{4}{k} \left( \frac{9}{19} \right)^k \left( \frac{10}{19} \right)^{4-k} \), where \(k\) is the number of wins.

03

Calculate Probability of Breaking Even

Breaking even occurs when you win exactly 2 out of 4 times. Using the binomial probability formula from Step 2, we calculate \( P(X = 2) = \binom{4}{2} \left( \frac{9}{19} \right)^2 \left( \frac{10}{19} \right)^2 \). Calculating this gives: \( P(X = 2) = 6 \times \left( \frac{81}{361} \right) \times \left( \frac{100}{361} \right) \approx 0.324 \).

04

Calculate Probability of Losing Money

Losing money occurs if you win on fewer than 2 plays, i.e., \(X = 0\) or \(X = 1\). Using the binomial distribution: \( P(X = 0) = \binom{4}{0} \left( \frac{9}{19} \right)^0 \left( \frac{10}{19} \right)^4 \) and \( P(X = 1) = \binom{4}{1} \left( \frac{9}{19} \right)^1 \left( \frac{10}{19} \right)^3 \). Calculating these, we find \( P(X < 2) = P(X = 0) + P(X = 1) \approx 0.178 + 0.356 = 0.534 \).

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Key Concepts

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

Roulette

Roulette is a popular casino game that has a rich history and is simple to understand. It consists of a spinning wheel with numbered slots and a small ball that the dealer rolls along the wheel in the opposite direction. The wheel includes 38 slots in American roulette, with numbers ranging from 0, 00, and 1 to 36. These slots are divided into red, black, and green colors:

• 18 red slots
• 18 black slots
• 2 green slots (0 and 00)

When playing roulette, players can choose to bet on a specific number, a group of numbers, or the color. Betting on red, for instance, means you are wagering that the ball will land in one of the 18 red slots. Given the mechanics of the roulette wheel and ball, each slot has an equal chance of being the landing spot, making roulette both a game of chance and strategy.

Binomial Distribution

The binomial distribution is a concept in probability and statistics that models the number of successes in a fixed number of independent trials. Each trial results in either success or failure, and the probability of success remains constant. This distribution is ideal for problems like betting on red in roulette.

For our roulette scenario, imagine you bet on red four times. Each bet is an independent trial with two possible outcomes: either the ball lands on red (a win) or it doesn't (a loss). Since there are a fixed number of trials (4 bets), and the probability of winning on red is constant (approximately 0.474), we can use the binomial distribution to predict how many times you win. Here, the parameters are:

• Number of trials (\(n\)): 4
• Probability of winning (\(p\)): \(\frac{9}{19}\)

This distribution helps calculate the likelihood of different outcomes, such as winning exactly two out of four bets.

Betting Strategy

Developing a betting strategy in roulette involves understanding probabilities and setting personal limits on betting amounts and outcomes. While no strategy can guarantee success because of the inherent randomness, understanding your expected outcomes, like breaking even or losing money, can help manage risks.

A basic strategy when betting on red is to understand what it means to 'break even.' In this context, you recover your initial investment by winning exactly two out of four bets. By knowing beforehand the probability of this outcome \(\approx 0.324\), you can decide whether this level of risk matches your personal betting goals. Additionally, being aware of the probability of losing money, which is when you win fewer than two bets \(\approx 0.534\), can guide decisions on when to stop, how much to bet, or whether to change strategies.

Probability Mass Function

The Probability Mass Function (PMF) is a fundamental concept for discrete random variables like the number of times you win in roulette betting. It provides a way to calculate the probability of each possible outcome, helping you understand the likelihood of different results when you bet on red multiple times.

In this exercise, the PMF is used to calculate probabilities of winning 0, 1, 2, 3, or 4 times when betting on red four times. Using the PMF for a binomial distribution, the probability of winning \(k\) times is given by:

• \(P(X = k) = \binom{4}{k} \left(\frac{9}{19}\right)^k \left(\frac{10}{19}\right)^{4-k}\)
• Here, \(\binom{4}{k}\) is the binomial coefficient, \(\left(\frac{9}{19}\right)^k\) is the probability of winning \(k\) times, and \(\left(\frac{10}{19}\right)^{4-k}\) is the probability of losing the remaining times.

This function is crucial for determining not just the average number of wins, but also the variability of your results, thus serving as a tool in evaluating betting strategies in games like roulette.