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Chapter 15: Problem 28

Roulette. A roulette wheel has 38 slots, of which 18 are black, 18 are red, and two are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. One of the simplest wagers is to choose red or black. A bet of \(\$ 1\) on red returns \(\$ 2\) if the ball lands in a red slot. Otherwise, the player loses the dollar. When gamblers bet on red or black, the two green slots result in losses. Because the probability of winning \(\$ 2\) is \(18 / 38\), the mean payoff from a \(\$ 1\) bet is twice \(18 / 38\), or \(94.7\) cents. Explain what the law of large numbers tells us about what will happen if a gambler makes very many bets on red.

Short Answer

Expert verified
As bets on red increase, average loss approaches 57.9 cents per bet.

Step by step solution

01

Understanding the Law of Large Numbers

The Law of Large Numbers states that as the number of experiments increases, the experimental or observed probability of an event approaches the theoretical probability. In the context of roulette, it means that the more bets you make, the closer your average payoff will get to the expected value.
02

Calculating the Expected Value for One Bet on Red

In roulette, betting on red means the probability of winning is \( \frac{18}{38} \). If you win, you receive \\(2 (your \\)1 bet and \\(1 gain), but if you lose, you lose your \\)1 bet. The expected value \( E \) of the bet can be calculated as:\[ E = \left( \frac{18}{38} \times 2 \right) + \left( \frac{20}{38} \times (-1) \right) \]Where the \( \frac{20}{38} \) represents the probability of not landing on red (18 black + 2 green slots).
03

Performing the Calculation

Substitute and simplify the expression:\[ E = \left( \frac{18}{38} \times 2 \right) + \left( \frac{20}{38} \times (-1) \right) = \left( \frac{36}{38} \right) - \left( \frac{20}{38} \right) = \frac{36 - 20}{38} = \frac{16}{38} \approx 0.421 \]So the expected payoff from a single \\(1 bet is approximately \\)0.421, indicating a loss on average of \$0.579 per bet.
04

Interpreting the Result with the Law of Large Numbers

The expected value \( 0.421 \) or 42.1 cents means that over a large number of spins, the average amount won per \\(1 bet on red will approach 42.1 cents. Consequently, since the player originally bets \\)1, their average loss will converge to about 57.9 cents per bet, given the house's edge.

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Key Concepts

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

Expected Value
When betting on events like roulette, expected value helps you understand the average outcome over the long run. It combines the probabilities of potential outcomes with their associated payoffs.
In roulette, if you bet $1 on red, your winning chances are determined by the event's probability:
  • The probability of winning (ball lands on red): \( \frac{18}{38} \)
  • If you win, you receive \(2 \) (your bet plus winnings)
  • The probability of losing (ball lands on black or green): \( \frac{20}{38} \)
  • If you lose, you lose \(1 \)
The expected value \( E \) can be calculated as follows:\[ E = \left( \frac{18}{38} \times 2 \right) + \left( \frac{20}{38} \times (-1) \right) \]This formula combines the winnings probability weighted by the gain and loss probability weighted by the loss. Calculating this gives you 42.1 cents, meaning you'll lose on average 57.9 cents per bet.
Probability in Roulette
Roulette offers a simple yet interesting probability scenario. When you place a bet on the color red:
  • There are 18 red slots out of a total of 38 on the wheel
  • Thus, the probability that the ball lands on red is \( \frac{18}{38} \) or approximately 47.4%
  • The probability it lands on any color other than red (black or green) is \( \frac{20}{38} \)
The probability of any event in roulette is based on the number of favorable outcomes divided by the total number of possible outcomes.
Though highly unlikely on single spins, over many, the chances resemble theoretical predictions, reinforcing the setup's fairness and predictability yet showing a player's probable loss due to the house edge.
Gambling Odds
Odds in gambling indicate the likelihood of a certain outcome occurring and are often expressed as a ratio. In roulette, odds offer insights into returns and risks:
  • For red, the betting odds favor the house slightly
  • Each red bet pits 18 red slots against the remaining 20 slots (black or green)
  • The loss odds are higher, reflecting the casino's built-in edge
The house edge is a key concept that players must understand, as it represents the average gross profit the casino expects to make from each game.
Roulette's house edge comes from its two green slots, which skew the otherwise even distribution of 18 black and 18 red.
Statistical Analysis in Gambling
Statistical analysis provides players insights into what to expect over time. It aggregates individual game data to form broader understandings of likelihoods and outcomes:
  • By analyzing expected value, players see potential long-term losses
  • Probability helps them anticipate event outcomes
  • Understanding odds lets players gauge best possible outcomes
Overall, statistical analysis in gambling helps in minimizing risks by enabling players to make more informed decisions, though it cannot alter the innate probabilities or house edges set by game designs.
With tools like these, serious bettors can mitigate some financial losses while amplifying their roulette strategy insights.

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Most popular questions from this chapter

Playing the Numbers: The House Has a Business. Unlike Joe (see the previous exercise), the operators of the numbers racket can rely on the law of large numbers. It is said that the New York City mobster Casper Holstein took as many as 25,000 bets per day in the Prohibition era. That's 150,000 bets in a week if he takes Sunday off. Casper's mean winnings per bet are \(\$ 0.40\) (he pays out an average of 60 cents per dollar bet to people like Joe and keeps the other 40 cents). His standard deviation for single bets is about \(\$ 18.96\), the same as Joe's. a. What are the mean and standard deviation of Casper's average winnings \(x\) on his 150,000 bets? b. According to the central limit theorem, what is the approximate probability that Casper's average winnings per bet are between \(\$ 0.30\) and \(\$ 0.50\) ? After only a week, Casper can be pretty confident that his winnings will be quite close to \(\$ 0.40\) per bet.

Annual returns on stocks vary a lot. The long-term mean return on stocks in the S\&P 500 is \(9.8 \%\), and the long-term standard deviation of returns is \(16.8 \%\). The law of large numbers says that a. you can get an average return higher than the mean \(9.8 \%\) by investing in a large number of the \(\mathrm{S} \& \mathrm{P}\) stocks. b. as you invest in more and more stocks chosen at random, your long-term average return on these stocks gets ever closer to \(9.8 \%\). c. if you invest in a large number of stocks chosen at random, your long-term average return will have approximately a Normal distribution.

Statistics Anxiety. What can teachers do to alleviate statistics anxiety in their students? To explore this question, statistics anxiety for students in two classes was compared. In one class, the instructor lectured in a formal manner, including dressing formally. In the other, the instructor was less formal, dressed casually, was more personal, used humor, and called on students by their first names. Anxiety was measured using a questionnaire. Higher scores indicate a greater level of anxiety. The mean anxiety score for students in the formal lecture class was 25.40; in the informal class, the mean was 20.41. For each of the boldface numbers, indicate whether it is a parameter or a statistic. Explain your answer.

The Bureau of Labor Statistics announces that last month it interviewed all members of the labor force in a sample of 60,000 households; \(3.5 \%\) of the people interviewed were unemployed. The boldface number is a a. sampling distribution. b. statistic. c. parameter.

What Does the Central Limit Theorem Say? Asked what the central limit theorem says, a student replies, "As you take larger and larger samples from a population, the histograms of the sample values look more and more Normal." Is the student right? Explain your answer.
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