Probability and Statistics > Basic Statistics > Expected Value
Contents
- What is Expected Value?
- Find an Expected Value by Hand
- Find an Expected Value in Excel
- Find an Expected Value for a Discrete Random Variable
- What is an Expected Value used for in Real Life?
What is Expected Value?
Expected value is exactly what you might think it means intuitively: the return you can expect for some kind of action, like how many questions you might get right if you guess on a multiple choice test.
For example, if you take a 20 question multiple-choice test with A,B,C,D as the answers, and you guess all “A”, then you can expect to get 25% right (5 out of 20). The math behind this kind of expected value is:
The probability (P) of getting a question right if you guess: .25
The number of questions on the test (n)*: 20
P x n = .25 x 20 = 5
*You might see this as X instead.
This type of expected value is called an expected value for a binomial random variable. It’s binomial because there are only two possible outcomes: you get the answer right, or you get the answer wrong.
Formula
The expected value formula for binomial random variables is written as E(X)=n*P (or P*n, X*P or P*X).
The expected value formula changes a little if you have a series of trials (for example, a series of coin tosses). When you have a series of trials, you take your basic formula (n*P) for each trial and then add them together. Mathematically, the expected value formula for a series of binomial trials is:
E(X) = x1P1 + x2P2 + x3P3 + . . . + xnPn.
The “…+ xnPn” just means to keep on adding your results together — even if you have 1,000 results you would still keep on adding.
If you make a chart, the math behind finding an expected value becomes clearer.
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Calculate an Expected value in statistics by hand
This section explains how to figure out the expected value for a single item (like purchasing a single raffle ticket) and what to do if you have multiple items. If you have a discrete random variable, read this other article instead: Expected value for a discrete random variable.
Sample question: You buy one $10 raffle ticket for a new car valued at $15,000. Two thousand tickets are sold. What is the expected value of your gain?
Step 1: Make a probability chart (see: How to construct a probability distribution). Put Gain(X) and Probability P(X) heading the rows and Win/Lose heading the columns.

Step 2: Figure out how much you could gain and lose. In our example, if we won, we’d be up $15,000 (less the $10 cost of the raffle ticket). If you lose, you’d be down $10. Fill in the data (I’m using Excel here, so the negative amounts are showing in red).

Step 3: In the bottom row, put your odds of winning or losing. Seeing as 2,000 tickets were sold, you have a 1/2000 chance of winning. And you also have a 1,999/2,000 probability chance of losing.

Step 4: Multiply the gains (X) in the top row by the Probabilities (P) in the bottom row.
$14,990 * 1/2000 = $7.495,
(-$10)*(1,999/2,000)= -$9.995
Step 5:Add the two values together:
$7.495 + -$9.995 = -$2.5.
That’s it!
Note on multiple items: for example, what if you purchase a $10 ticket, 200 tickets are sold, and as well as a car, you have runner up prizes of a CD player and luggage set?
Perform the steps exactly as above. Make a probability chart except you’ll have more items:

Then multiply/add the probabilities as in step 4: 14,990*(1/200) + 100 * (1/200) + 200 * (1/200) + -$10 * (197/200).
You’ll note now that because you have 3 prizes, you have 3 chances of winning, so your chance of losing decreases to 197/200.
Note on the formula: The actual formula for expected gain is E(X)=∑X*P(X) (this is also one of the AP Statistics formulas). What this is saying (in English) is “The expected value is the sum of all the gains multiplied by their individual probabilities.”
Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step explanations, just like this one!
Find an Expected Value in Excel

Step 1: Type your values into two columns in Excel (“x” in one column and “f(x)” in the next.
Step 2: Click an empty cell.
Step 3: Type =SUMPRODUCT(A2:A6,B2:B6) into the cell where A2:A6 is the actual location of your x variables and f(x) is the actual location of your f(x) variables.
Step 4: Press Enter.
That’s it!
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Find an Expected Value for a Discrete Random Variable
You can think of an expected value as a mean, or average, for a probability distribution. A discrete random variable is a random variable that can only take on a certain number of values. For example, if you were rolling a die, it can only have the set of numbers {1,2,3,4,5,6}. The expected value formula for a discrete random variable is:

Basically, all the formula is telling you to do is find the mean by adding the probabilities. The mean and the expected value are so closely related they are basically the same thing. You’ll need to do this slightly differently depending on if you have a set of values, a set of probabilities, or a formula.
Expected Value Discrete Random Variable (given a list).
Sample problem #1: The weights (X) of patients at a clinic (in pounds), are: 108, 110, 123, 134, 135, 145, 167, 187, 199. Assume one of the patients is chosen at random. What is the EV?
Step 1: Find the mean. The mean is:
108 + 110 + 123 + 134 + 135 + 145 + 167 + 187 + 199 = 145.333.
That’s it!
Expected Value Discrete Random Variable (given “X”).
Sample problem #2. You toss a fair coin three times. X is the number of heads which appear. What is the EV?
Step 1: Figure out the possible values for X. For a three coin toss, you could get anywhere from 0 to 3 heads. So your values for X are 0,1,2 and 3.
Step 2: Figure out your probability of getting each value of X. You may need to use a sample space (The sample space for this problem is: {HHH TTT TTH THT HTT HHT HTH THH}). The probabilities are: 1/8 for 0 heads, 3/8 for 1 head, 3/8 for two heads, and 1/8 for 3 heads.
Step 3: Multiply your X values in Step 1 by the probabilities from step 2.
E(X) = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8) = 3/2.
The EV is 3/2.
Expected Value Discrete Random Variable (given a formula, f(x)).
Sample problem #3. You toss a coin until a tail comes up. The probability density function is f(x) = 1/2x. What is the EV?
Step 1: Insert your “x” values into the first few values for the formula, one by one. For this particular formula, you’ll get:
1/20 + 1/21 + 1/22 + 1/23 + 1/24 + 1/25.
Step 2: Add up the values from Step 1:
= 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 1.96875.
Note: What you are looking for here is a number that the series converges on (i.e. a set number that the values are heading towards). In this case, the values are headed towards 2, so that is your EV.
Tip: You can only use the expected value discrete random variable formula if your function converges absolutely. In other words, the function must stop at a particular value. If it doesn’t converge, then there is no EV.
What is Expected Value in Statistics used for in Real Life?
Expected values for binomial random variables (i.e. where you have two variables) are probably the simplest type of expected values. In real life, you’re likely to encounter more complex expected values that have more than two possibilities. For example, you might buy a scratch off lottery ticket with prizes of $1000, $10 and $1. You might want to know what the payoff is going to be if you go ahead and spend $1, $5 or even $25.
Let’s say your school is raffling off a season pass to a local theme park, and that value is $200. If the school sells one thousand $10 tickets, every person who buys the ticket will lose $9.80, expect for the person who wins the season pass. That’s a losing proposition for you (although the school will rake it in). You might want to save your money! Here’s the math behind it:
- The value of winning the season ticket is $199 (you don’t get the $10 back that you spent on the ticket.
- The odds that you win the season pass are 1 out of 1000.
- Multiply (1) by (2) to get: $199 * 0.001 = 0.199. Set this number aside for a moment.
- The odds that you lose are 999 out of 1000. In other words, your odds of ending up minus ten dollars are 999/1000. Multiplying -$10, you get -9.999.
- Adding (3) and (4) gives us the expected value: 0.199 + -9.999 = -9.80.
Here’s that scenario in a table:

Related articles:
Online expected value calculator
If you’re looking for more information on formula variations (this gets a bit more technical!), see this article at Wolfram.
If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you’re are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.
The way that this seems to be is that you need to know how to set up your tables with the information given to you. And this is where I am seeing were I am having problems, what goes where and why? I am having problems with that formula E(X)=Ex*P(X)I really don’t understand it. I guess if I go back to where this started and re-read it the section maybe I will get the jest of it. I see how they put the tables together thats not hard its just trying to figure out where the information goes.
Lisa,
If you follow the steps in this how-to, you can skip using the formula. But you are right–it’s a matter of figuring out where to put the information, which is sometimes a challenge. I’m sure with practice you’ll pick it up,
Stephanie
I agree with Lisa. I am having a hard time understanding where the information goes. This explanation does help a little, I guess I just need to do it more often.
I too agree, sometimes the biggest challenge is to know where to plug in the numbers in the equation. The more problems I practice, the more it seems to click, though. Your explanations on here are clear cut and easy to follow.
This blog really helped me figure out probability charts. I agree with the other post that it was hard to figure out at first, but after practicing over and over it finally came to me. I also like that it shows the possibility of winning multiple prizes. I don’t recall the book having an example like this one.
the examples are so helpful when you make tables. Without making the tables, it gets confusing. The more examples the better.
Actually, I believe the formula is E(x) = (-losses * P(losing)) + (gains * P(gains)).
I’m a derp and i don’t understand
Expected Value Discrete Random Variable (given “X”).
Sample problem #2. You toss a fair coin three times. X is the number of heads which appear. What is the EV?
Step 1: Figure out the possible values for X. For a three coin toss, you could get anywhere from 0 to 3 heads. So your values for X are 0,1,2 and 3.
Step 2: Figure out your probability of getting each value of X. You may need to use a sample space. The probabilities are: 1/8 for 0 heads, 3/8 for 1 head, 3/8 for two heads, and 1/8 for 3 heads.
Step 3: Multiply your X values in Step 1 by the probabilities from step 2.
E(X) = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8) = 3/2.
The EV is 3/2.
question how are you computing step 2 to get 1/8 for 0 heads, 3/8 for 1 head, 3/8 for two heads and 1/8 for 3 heads?
The sample space for this problem is: {HHH TTT TTH THT HTT HHT HTH THH}