# Probability

## Expected Value


### Learning Objectives

After completing this section, you should be able to:
1. Calculate the expected value of an experiment.
2. Interpret the expected value of an experiment.
3. Use expected value to analyze applications.
The casino game roulette has dozens of different bets that can be made. These bets have different probabilities of winning but also have different payouts. In general, the lower the probability of winning a bet is, the more money a player wins for that bet. With so many options, is there one bet that’s “smarter” than the rest? What’s the best play to make at a roulette table? In this section, we’ll develop the tools we need to answer these questions.

### Expected Value

Many experiments have numbers associated with their outcomes. Some are easy to define; if you roll 2 dice, the sum of the numbers showing is a good example. In some card games, cards have different point values associated with them; for example, in some forms of the game rummy, aces are worth 15 points; 10s, jacks, queens, and kings are worth 10; and all other cards are worth 5. The outcomes of casino and lottery games are all associated with an amount of money won or lost. These outcome values are used to find the expected value of an experiment: the mean of the values associated with the outcomes that we would observe over a large number of repetitions of the experiment. (See Conditional Probability and the Multiplication Rule for more on means.)
That definition is a little vague; How many is “a large number?” In practice, it depends on the experiment; the number has to be large enough that every outcome would be expected to appear at least a few times. For example, if we’re talking about rolling a standard 6-sided die and we note the number showing, a few dozen replications should be enough that the mean would be representative. Since the probability of each outcome is , we would expect to see each outcome about 8 times over the course of 48 replications. However, if we’re talking about the Powerball lottery, where the probability of winning the jackpot is about , we would need several billion replications to ensure that every outcome appears a few times. Luckily, we can find the theoretical expected value before we even run the experiment the first time.
Let’s make note of some things we can learn from . First, as Exercises 1 and 3 demonstrate, the expected value of an experiment might not be a value that could come up in the experiment. Remember that the expected value is interpreted as a mean, and the mean of a collection of numbers doesn’t have to actually be one of those numbers.
 Second, looking at Exercise 1, the expected value (3.5) was just the mean of the numbers on the faces of the die: . This is no accident! If we break that fraction up using the addition in the numerator, we get , which we can rewrite as . That’s exactly the computation we did to find the expected value! In fact, expected values can always be treated as a special kind of mean called a weighted mean, where the weights are the probabilities associated with each value. When the probabilities are all equal, the weighted mean is just the regular mean.

### Interpreting Expected Values

As we noted, the expected value of an experiment is the mean of the values we would observe if we repeated the experiment a large number of times. (This interpretation is due to an important theorem in the theory of probability called the Law of Large Numbers.) Let’s use that to interpret the results of the previous example.

### Using Expected Value

Now that we know how to find and interpret expected values, we can turn our attention to using them. Suppose someone offers to play a game with you. If you roll a die and get a 6, you get $10. However, if you get a 5 or below, you lose $1. Is this a game you’d want to play? Let’s look at the expected value: The probability of winning is  and the probability of losing is , so the expected value is . That means, on average, you’ll come out ahead by about 83 cents every time you play this game. It’s a great deal! On the other hand, if the winnings for rolling a 6 drop to $3, the expected value becomes , meaning you should expect to lose about 33 cents on average for every time you play. Playing that game is not a good idea! In general, this is how casinos and lottery corporations make money: Every game has a negative expected value for the player.

### Check Your Understanding


### Key Terms

1. expected value

### Key Concepts

1. The expected value of an experiment is the sum of the products of the numerical outcomes of an experiment with their corresponding probabilities.
2. The expected value of an experiment is the most likely value of the average of a large number of replications of the experiment.

### Formulas

1. If  represents an outcome of an experiment and  represents the value of that outcome, then the expected value of the experiment is:
    
    where 
    

### Projects

1. The Binomial Distribution is one of many examples of a discrete probability distribution. Other examples include the Geometric, Hypergeometric, Multinomial, Poisson, and Negative Binomial Distributions. Choose one of these distributions, and find out what makes it different from the Binomial Distribution. In what situations can it be applied? How is it used? Once you have an idea of how it’s used, write a series of five questions like the ones in this chapter that can be answered with that distribution, and find the answers.
2. Binomial is a word that also comes up in algebra; the word describes polynomials with two terms. At first glance, there isn’t much to indicate that these two uses of the word are related, but it turns out there is a connection. Explore the connection between the Binomial Distribution and the algebraic concept of binomial expansion, (the process of multiplying out expressions like  for a positive whole number ). Search for a connection with the mathematical object known as Pascal’s Triangle.
3. Hazard is a dice game that was mentioned in Chaucer’s Canterbury Tales. It was a popular game of chance played in taverns and coffee houses well into the 18th century; its popularity at the time of the foundation of probability theory means that it was a common example in early texts on finding expected values and probabilities. Find the rules of the game, and get some practice playing it. Then, analyze the choices that the caster gets to make, and decide which is most advantageous, using the language of expected values.

### Chapter Review


### The Multiplication Rule for Counting


### Permutations


### Combinations


### Tree Diagrams, Tables, and Outcomes


### Basic Concepts of Probability


### Probability with Permutations and Combinations


### What Are the Odds?


### The Addition Rule for Probability


### Conditional Probability and the Multiplication Rule


### The Binomial Distribution


### Expected Value


### Chapter Test
