Problem 18 - Algebra and Trigonometry

Question

Slot Machine A slot machine has three wheels, and each wheel has 11 positions: the digits \(0,1,2, \ldots, 9\) and the picture of a watermelon. When a quarter is placed in the machine and the handle is pulled, the three wheels spin independently and come to rest. When three watermelons show, the payout is \(\$ 5 ;\) otherwise, nothing is paid. What is the expected value of this game?

Step-by-Step Solution

Answer

The expected value of the game is approximately \(-\$0.246\).

Step 1: Understand the Probability of Winning

Each wheel has 11 positions: digits 0 through 9 and one watermelon. The probability that a wheel stops on a watermelon is \(\frac{1}{11}\). Since the wheels operate independently, the probability of all three wheels stopping on a watermelon (winning) is \(\left(\frac{1}{11}\right)^3 = \frac{1}{1331}\).

Step 2: Determine the Probability of Losing

If not all the wheels show watermelons, the outcome is a loss. The probability of a wheel not showing a watermelon is \(1 - \frac{1}{11} = \frac{10}{11}\). Thus, the probability of all three wheels not showing watermelons, resulting in a loss, is \(1 - \frac{1}{1331}\).

Step 3: Calculate the Expected Value of Winning and Losing

For winning, you get \(\\(5\) with the probability of \(\frac{1}{1331}\). Thus, the expected value of winning is \(5 \times \frac{1}{1331} = \frac{5}{1331}\). For losing, the payout is \(\\)0\), so no expected value contribution here, besides the initial stake lost.

Step 4: Compute the Total Expected Value

The expected value (EV) of the game is the sum of the expected values from winning and losing. However, include the cost to play, which is \(\$0.25\) per game. Thus, the EV is \(\frac{5}{1331} - 0.25\).

Step 5: Simplify the Expected Value

To find the EV in more understandable terms, convert \(\$0.25\) into a fraction over 1331. Since \(0.25 = \frac{1}{4}\), this is equivalent to multiplying \(\frac{1}{4}\) by \(\frac{1331}{1331}\) to get \(\frac{332.75}{1331}\). Thus, \(EV = \frac{5}{1331} - \frac{332.75}{1331} = \frac{5 - 332.75}{1331} = \frac{-327.75}{1331}\).

Step 6: Interpret the Expected Value

The expected value \(\frac{-327.75}{1331}\) is a negative value. This means that on average, for every \(\\(0.25\) bet, the player loses about \\)0.246 per game (once decimal approximation is made).

Key Concepts

Probability

Probability is a measure of the likelihood that a certain event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In the context of the slot machine, each wheel landing on the watermelon has a probability of \(\frac{1}{11}\). This is because there is only one watermelon on each wheel out of 11 possible positions.

To win the game, all three wheels must show the watermelon. Since each wheel operates independently, the probability of winning is calculated by multiplying the probabilities of each wheel landing on the watermelon: \(\left(\frac{1}{11}\right)^3 = \frac{1}{1331}\). This shows a very low chance of winning, reflecting the rarity of the event.

The concept of probability helps us understand and predict the outcomes of random processes like betting on a slot machine. It also guides us in calculating more complex probabilities based on simpler events.

Independent Events

Independent events are events whose outcomes do not affect each other. This means the occurrence of one event does not change the probability of the other event occurring. In the slot machine problem, each wheel spins independently, meaning that the stopping position of one wheel has no effect on the others.

This independence is crucial for calculating probabilities. For the wheels to all show watermelons, the independence allows us to multiply the probabilities of each wheel stopping on a watermelon: \(\frac{1}{11} \times \frac{1}{11} \times \frac{1}{11} = \frac{1}{1331}\).

Understanding independent events helps in grasping how combined probabilities are calculated, especially in games of chance where multiple outcomes must align to achieve a desired result.

Random Variable

A random variable is a numerical outcome of a random process, and it is often denoted by a symbol like \(X\). For the slot machine, the random variable could be defined as the payout a player receives.

Random variables can be helpful in modeling and analyzing scenarios, such as calculating expected values. In the slot machine example, the random variable takes two values: \(5\) (the winning payout) with probability \(\frac{1}{1331}\), and \(0\) (the losing payout) with probability \(1 - \frac{1}{1331}\).

By assigning values to the possible outcomes, random variables enable us to perform statistical analysis and make predictions about expected gains and losses over time.

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. It is essentially a list of the probabilities associated with each of the possible outcomes of a random variable.

In the slot machine scenario, the probability distribution is quite simple. We have two outcomes: winning \(\\(5\) with a probability of \(\frac{1}{1331}\) and losing \(\\)0\) with a probability of \(1 - \frac{1}{1331}\). This distribution helps us compute the expected value of the game.

Understanding a probability distribution allows us to evaluate the likelihood of different outcomes and their associated impacts. It is an essential tool in both theoretical and applied statistics, allowing us to understand and quantify risk and reward in games of chance like this one.