Expected Values (DP IB Analysis & Approaches (AA)): Revision Note

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 with its corresponding probability

  • Adding all these terms together

    
    
    

    • This is given in the exam formula booklet

• Look out for symmetrical distributions (where the values of X are symmetrical and their probabilities are symmetrical)

  • 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 positive if there is a gain, and 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

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