Expected Values (DP IB Applications & Interpretation (AI)) : Revision Note

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Expected Values E(X)

What does E(X) mean and how do I calculate E(X)?

  • E(X) means the expected value or the mean of a random variable X

    • The expected value does not need to be an obtainable value of X

    • For example: the expected value number of times a coin will land on tails when flipped 5 times is 2.5

  • For a discrete random variable, it is calculated by:

    • Multiplying each value of X with its corresponding probability

    • Adding all these terms together


      begin mathsize 22px style straight E left parenthesis X right parenthesis equals sum x straight P left parenthesis X equals x right parenthesis end style

      • This is given in the formula booklet

  • Look out for symmetrical distributions (where the values of X are symmetrical and their probabilities are symmetrical) as the mean of these is the same as the median

    • For example: if X can take the values 1, 5, 9 with probabilities 0.3, 0.4, 0.3 respectively then by symmetry the mean would be 5

How can I decide if a game is fair?

  • Let X be the random variable that represents the gain/loss of a player in a game

    • X will be negative if there is a loss

  • Normally the expected gain or loss is calculated by subtracting the cost to play the game from the expected value of the prize

  • If E(X) is positive then it means the player can expect to make a gain

  • If E(X) is negative then it means the player can expect to make a loss

  • The game is called fair if the expected gain is 0

    • E(X) = 0

Worked Example

Daphne pays $15 to play a game where she wins a prize of $1, $5, $10 or $100. The random variable W represents the amount she wins and has the probability distribution shown in the following table:

w

1

5

10

100

straight P left parenthesis W equals w right parenthesis

0.35

0.5

0.05

0.1

a) Calculate the expected value of Daphne's prize.

4-4-2-ib-ai-aa-sl-expected-we-a-solution

b) Determine whether the game is fair.

4-4-2-ib-ai-aa-sl-expected-we-b-solution
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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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