How to figure out an expected value in statistics

binomial distribution 8 Comments

Like most phrases you’ll come across in probability and statistics, expected value sounds like a daunting undertaking. Until you realize it only takes a few simple steps to figure it out. This article explains how to figure out the expected value for a single item (like purchasing a single raffle ticket). Further down in the page you’ll find an explanation of what to do if you have multiple items.

Expected value in statistics

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: Construct a probability chart. Put Gain(X) and Probability P(X) heading the rows and Win/Lose heading the columns.
expected value

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).

expected value 2

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.

expected value 3

Step 4: Multiply the gains (X) in the top won by the Probabilities (P) in the bottom row.
$14,990 * 1/2000 = 7.495,
(-$10)*(1,999/2,000)= -$9.995

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:

expected value 4

Then multiply/add the probababilities 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 in probability and statistics is E(X)=∑X*P(X).  What this is saying (in English) is “The expected value is the sum of all the gains multiplied by their individual probabilities.”

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8 Responses to “How to figure out an expected value in statistics”

  1. Lisa Barcomb said:

    Sep 20, 09 at 4:09 am

    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.

  2. Stephanie said:

    Sep 20, 09 at 4:26 am

    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

  3. Mary Johnson said:

    Sep 23, 09 at 2:37 pm

    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.

  4. Donna Allen said:

    Sep 26, 09 at 7:35 am

    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.

  5. Catherine Flanagan said:

    Sep 27, 09 at 12:04 pm

    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.

  6. angie widdows said:

    Nov 03, 09 at 6:07 pm

    the examples are so helpful when you make tables. Without making the tables, it gets confusing. The more examples the better.

  7. Montana said:

    Jul 25, 10 at 12:49 pm

    Actually, I believe the formula is E(x) = (-losses * P(losing)) + (gains * P(gains)).

  8. Derp said:

    Apr 14, 11 at 7:06 am

    I’m a derp and i don’t understand

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