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The game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their color, they double their money. If it lands on another color, they lose their money. (a) Suppose you play roulette and bet $$\$ 3$$ on a single round. What is the expected value and standard deviation of your total winnings? (b) Suppose you bet $$\$ 1$$ in three different rounds. What is the expected value and standard deviation of your total winnings? (c) How do your answers to parts (a) and (b) compare? What does this say about the riskiness of the two games?

Step by step solution

01

Understand the Probabilities

In European roulette, there are 18 red slots, 18 black slots, and 1 green slot, making a total of 37 slots. The probability of landing on red is \( P(\text{red}) = \frac{18}{37} \), the probability of landing on black is \( P(\text{black}) = \frac{18}{37} \), and the probability of landing on green (neither red nor black, resulting in a loss) is \( P(\text{green}) = \frac{1}{37} \).

02

Step 2(a): Calculate Expected Winnings for a Single $3 Bet

When you bet \(3, you win \)3 if you win and lose all \(3 if you lose. Thus, the total outcome for winning is \(\)3\), and for losing is \(-$3\). The expected value (mean winnings) can be calculated as follows:\[ E(X_1) = (3) \cdot \frac{18}{37} + (-3) \cdot \left(1 - \frac{18}{37}\right) = \frac{18}{37} \times 6 - 3 \] Calculating the numbers, we find that:\[ E(X_1) = \frac{108}{37} - 3 \approx -0.1622 \]

03

Step 3(a): Calculate Standard Deviation for a Single $3 Bet

First compute the variance, which involves finding the squared differences between each outcome and the expectation. The variance is calculated as:\[ \text{Var}(X_1) = (3 - (-0.1622))^2 \cdot \frac{18}{37} + (-3 - (-0.1622))^2 \cdot \left(1 - \frac{18}{37} \right) \]This results in:\[ \text{Var}(X_1) = \frac{18}{37} \times (3.1622)^2 + \frac{19}{37} \times (2.8378)^2 \]And the standard deviation is the square root of the variance.

04

Step 4(b): Calculate Expected Winnings for Three $1 Bets in Separate Rounds

The expected value for one round remains the same as above calculated \(-0.1622\), but scaled for $1 bets:\[ E(X_2) = 1\cdot \left(\frac{18}{37} \times 2 + 0 - 1\right) = \frac{18}{37} \times 1 - 1 \]For one round, \[ E(X_2) = \frac{-0.0541}{3} \approx -0.1622 \] For three rounds, by linearity of expectation: \[ E(3X_2) = 3 \times (-0.0541) = -0.1623 \]

05

Step 5(b): Calculate Standard Deviation for Three $1 Bets in Separate Rounds

Each bet of \(1 is independent, so we use the variance for one \)1 bet and scale accordingly. The variance of a single $1 bet:\[ \text{Var}(X_2) = (1.1622)^2 \cdot \frac{18}{37} + (1.8378)^2 \cdot \left(1 - \frac{18}{37} \right) \]And the total variance for three rounds is: \[ \text{Var}(3X_2) = 3 \times \text{Var}(X_2) \] The standard deviation is then the square root of this variance.

06

Step 6(c): Compare Expected Values and Standard Deviations

In both scenarios, the expected value of the total winnings is \(-0.1622\), indicating that either strategy leads to similar expected losses. However, the standard deviation for a single \(3 bet will be greater than that for three separate \)1 bets because variance is additive and thus larger spread for three independent trials reduces overall uncertainty.

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Key Concepts

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

Expected Value

One of the most powerful concepts in probability and statistics is the **expected value**. In simple terms, it is the average amount one can expect to win or lose per bet if the bet is repeated many times. It helps you understand the long-term trends in gambling or any random scenario.

• The expected value is calculated by summing up all possible outcomes, each multiplied by its probability of happening.
• In our European roulette scenario, if you bet $3, the expected value of your reasonable winnings is about -0.1622. This result is calculated using probabilities for red and black outcomes.
• An expected value of a negative number, such as -0.1622, indicates that, on average, you will lose that amount per bet in the long run.

Understanding expected value is crucial for making informed decisions when engaging in activities involving chance, such as casino games.

Standard Deviation

The **standard deviation** measures the variability or spread of potential outcomes around the expected value. It indicates how much individual bets might deviate from the average outcome.

• A higher standard deviation indicates a wide range of possible results, meaning more volatility in your gambling experience.
• For a single $3 roulette bet, the standard deviation is derived from the variance and involves calculating how far outcomes such as $3 or -$3 fall from the expected value.
• Understanding standard deviation can help determine the risk level of a gambling strategy, whether it involves large fluctuations or more consistent outcomes.

In gambling, knowing the standard deviation can give you insight into how unpredictable your betting results may be, helping you choose a strategy that aligns with your comfort level.

Variance

**Variance** is another essential statistical measure closely related to the standard deviation that measures how much the outcomes of a random variable, like a gambling bet, differ from the expected value.

• Variance is calculated by taking the average of the squared differences from the expected value, which depicts the spread of your potential winnings or losses.
• In the roulette problem, variance helps to quantify the total risk for multiple $1 bets across different rounds.
• A greater variance indicates a larger spread of outcomes and, consequently, a riskier betting scenario.

By understanding variance and its impact on the standard deviation, you can better assess the volatility of a betting strategy and anticipate the range of your potential gambling results.

European Roulette

**European Roulette** is a popular casino game that involves betting on the outcome of a ball landing in one of the 37 slots of a spinning wheel.

• The wheel consists of 18 red slots, 18 black slots, and 1 green slot (zero), and the odds of landing on each color are equal for each spin.
• Unlike its American counterpart, which includes 38 slots with two green zeros, European Roulette gives slightly better odds due to having fewer slots.
• Roulette is a game of chance with calculated probabilities, which allows players to estimate their potential losses or gains using expected value and standard deviation.

Playing European Roulette involves understanding these odds and using them to make educated bets, balancing potential rewards against the likelihood of each outcome.