Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

a. Show that, for any number \(c\), $$E(c X)=c E(X)$$ b. Use this result to find the expected loss if a gambler bets \(\$ 300\) on \(\mathrm{red}\) in a single play in American roulette.

Step-by-Step Solution

Answer

We have shown that for any number \(c\), \(E(cX) = cE(X)\). Now, for the American roulette problem, we consider the random variable X as the net gain in a single play. There are 38 slots, with 18 red, 18 black, and 2 green. The probabilities of winning or losing are \(\frac{18}{38}\) and \(\frac{20}{38}\), respectively. Using the expected value formula and the given bet amount of $300, the expected loss in a single play is approximately -\$15.79.

Step 1: a. Demonstrating Expectation of multiple of a Random Variable

The expected value of a random variable is given by: $$E(X) = \sum_{i=1}^{n} x_iP(x_i)$$ where \(x_i\) are the possible outcomes and \(P(x_i)\) are their corresponding probabilities. Now, let us find the expected value of the constant times the random variable: $$E(cX) = \sum_{i=1}^{n} (cx_i) P(x_i)$$ We can factor the constant \(c\) out of the summation, since it does not change for each event: $$E(cX) = c\sum_{i=1}^{n} x_iP(x_i)$$ Comparing this expression with the expression for \(E(X)\), we can see that: $$E(cX) = cE(X)$$ Thus, we have shown that the expected value of a random variable multiplied by a constant is equal to the constant times the expected value of the random variable.

Step 2: b. Calculating the Expected Loss in American Roulette

In American roulette, there are 38 slots, 18 of which are red, 18 are black, and 2 are green. Since the gambler has bet $300 on red, we will consider this random variable: X = the net gain in a single play Possible outcomes: - win: +300 (if the ball lands on red) - lose: -300 (if the ball lands on black or green) The probability of each outcome in American roulette: - P(win) = 18/38 (the probability of landing on red) - P(lose) = 20/38 (the probability of landing on black or green) Now we can use the expected value formula to find the expected loss (Note: In this context, a negative expected value represents an expected loss): $$E(X) = (+300) \times \frac{18}{38} + (-300) \times \frac{20}{38}$$ To find the expected loss, we calculate this expression: $$E(X) = 300 \times \frac{18 - 20}{38} = 300 \times \frac{-2}{38} = -\frac{300}{19} \approx -\$15.79$$ In a single play, the gambler's expected loss when betting \(300 on red in American roulette is approximately \)15.79.

Key Concepts

Random Variable

A random variable is a concept that represents numerical outcomes of random phenomena. Imagine rolling a die; the numbers 1 through 6 are outcomes, and together they form what we call a random variable. This concept is very important in probability and statistics because it helps us quantify uncertainty.
For each possible outcome, there is an associated probability that describes the likelihood of that outcome occurring. We often denote a random variable by symbols such as X or Y. For example, if X represents the number of heads when you flip a coin, then X can be 0 (if tails) or 1 (if heads).
Understanding random variables allows us to calculate expectations, variances, and other statistical measures that are foundational in probability theory.

Probability

Probability is a measure that quantifies the likelihood of a certain event occurring. It's a central idea in statistics and is used to predict how likely outcomes are to occur. The probability of an event is represented as a number between 0 and 1. A probability of 0 means the event will not happen, while a probability of 1 means it will certainly happen.
When considering complex systems or multiple events, probabilities can be combined using rules of probability, like the addition rule or multiplication rule. For example, the probability of rolling a 4 on a fair die is 1/6, because there is one favorable outcome over six possible outcomes.
In games of chance like roulette, probability helps you understand the odds of winning or losing, which is crucial for making informed decisions.

American Roulette

American Roulette is a popular casino game with a spinning wheel containing 38 slots. These slots are numbered 1 to 36, with additional slots labeled 0 and 00. The numbers 1 through 36 are split equally between red and black colors, while 0 and 00 are green.
When playing, you can place various types of bets, like betting on a single number, a range of numbers, or a color. The payout of the bet depends on its probability. In American Roulette, the presence of the two green slots (0 and 00) decreases the probability of winning bets compared to its European counterpart, which only includes one green slot (0).
The calculation of expected values, as demonstrated in gambling problems, gives insight into the fairness of the game. Knowing that the expected value of betting on a single color results in a loss, reflects the built-in house edge of the casino.

Gambling Mathematics

Gambling mathematics involves using mathematical techniques to understand and analyze games of chance. At its core is the concept of expected value, which helps determine the average outcome of repeated gambling actions over time.
By calculating expected values, gamblers can make more informed choices about their bets. For instance, if a player frequently makes bets where the expected value is negative, they are likely to incur losses in the long run.
Other elements of gambling mathematics include calculating odds, understanding variance, and using probability distributions to predict an outcome's fluctuations. By applying these principles, players can gauge whether their strategies might offer potential profits or only losses.