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Chapter 5: Problem 55

Explain the Law of Large Numbers. How does this law apply to gambling casinos?

Short Answer

Expert verified
The Law of Large Numbers ensures that over many trials, the average result converges to the expected value. Casinos use this principle to guarantee profit over the long term by relying on the house edge.

Step by step solution

01

Understand the Law of Large Numbers

The Law of Large Numbers (LLN) states that as the number of trials of a random experiment increases, the average of the results from those trials will approach the expected value. Mathematically, if you repeat an experiment many times, the sample mean will converge to the population mean.
02

The Formula

In mathematical terms, if you have a sequence of independent and identically distributed random variables, \(X_1, X_2, \ldots, X_n\), with expected value \(E[ X_i ] = \mu\), then the sample average \(\frac{1}{n} \sum_{i=1}^n X_i \) will converge to \mu\ as \(n\) becomes large.
03

Applying LLN to Gambling Casinos

In gambling, casinos rely on the Law of Large Numbers to ensure profitability. Each game has a built-in house edge, which means the casino expects to win a small percentage of all bets. While players may win or lose over the short term, over a large number of games, the average winnings for the casino will approach the expected house edge, guaranteeing profit in the long run.

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Key Concepts

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

Random Experiments
A random experiment is an action or process that leads to one of several possible outcomes, where the outcome is uncertain. For example, when you roll a die, you don't know what number will show up; hence, it's a random experiment. Understanding random experiments is crucial because these serve as the building blocks for many concepts in probability and statistics, including the Law of Large Numbers (LLN). When you repeat a random experiment multiple times, you can start to see patterns and make more accurate predictions.
Expected Value
The expected value is a core concept in probability and statistics. It is the average of all possible outcomes, weighted by their probabilities. Formally, if you have a random variable X that can take on values with probabilities, the expected value, E[X], is the sum of each possible value of X times its probability. The expected value gives you a long-term average result if you were to repeat an experiment many times. For example, in a gambling scenario, if a game has a 50% chance to win \(10 and a 50% chance to lose \)5, the expected value is calculated as:
  • 0.5 * \(10 + 0.5 * (-\)5) = \(2.50
This tells you that, on average, you can expect to win \)2.50 per game if you play an infinite number of times.
House Edge
The house edge is a term used in gambling to describe the advantage that the casino has over the players over the long run. It is expressed as a percentage of the player's bet that the casino expects to keep as profit. For instance, if a game has a house edge of 5%, this means that, on average, the casino expects to make a profit of 5% of all bets placed on that game.
  • The house edge ensures that, despite the randomness of individual outcomes, the casino will always make a profit over a large number of games.
  • Players may win or lose in the short term, but the Law of Large Numbers guarantees the casino profit in the long run.
Understanding the house edge is essential for gamblers, as it helps them realize that the odds are always tilted in favor of the casino.

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Most popular questions from this chapter

A National Ambulatory Medical Care Survey administered by the Centers for Disease Control found that the probability a randomly selected patient visited the doctor for a blood pressure check is \(0.593 .\) The probability a randomly selected patient visited the doctor for urinalysis is 0.064. Can we compute the probability of randomly selecting a patient who visited the doctor for a blood pressure check or urinalysis by adding these probabilities? Why or why not?

Suppose a mother already has three girls from three separate pregnancies. Does the fact that the mother already has three girls affect the likelihood of having a fourth girl? Explain.

A certain digital music player randomly plays each of 10 songs. Once a song is played, it is not repeated until all the songs have been played. In how many different ways can the player play the 10 songs?

In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, "When you see an ad emphasizing that a product is 'Made in America,' are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?" The results of the survey, by age group, are presented in the following contingency table. $$ \begin{array}{lrrrrr} & \mathbf{1 8 - 3 4} & \mathbf{3 5 - 4 4} & \mathbf{4 5 - 5 4} & \mathbf{5 5 +} & \text { Total } \\ \hline \text { More likely } & 238 & 329 & 360 & 402 & \mathbf{1 3 2 9} \\ \hline \text { Less likely } & 22 & 6 & 22 & 16 & \mathbf{6 6} \\ \hline \begin{array}{l} \text { Neither more } \\ \text { nor less likely } \end{array} & 282 & 201 & 164 & 118 & \mathbf{7 6 5} \\ \hline \text { Total } & \mathbf{5 4 2} & \mathbf{5 3 6} & \mathbf{5 4 6} & \mathbf{5 3 6} & \mathbf{2 1 6 0} \end{array} $$ (a) What is the probability that a randomly selected individual is 35-44 years of age, given the individual is more likely to buy a product emphasized as "Made in America"? (b) What is the probability that a randomly selected individual is more likely to buy a product emphasized as "Made in America," given the individual is \(35-44\) years of age? (c) Are 18 - to 34 -year-olds more likely to buy a product emphasized as "Made in America" than individuals in general?

How many different simple random samples of size 5 can be obtained from a population whose size is 50?
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