Expected Values (DP IB Maths: AI HL)

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Dan

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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

Author: Dan

Expertise: Maths

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.