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Fair/ Bias Expected Value A biased die is used in a game. The probabilities of getting the six different numbers on the die are shown in the table below. Expected Value E(X) is the average outcome of an event Number Probability πΈ π = π₯. π(π₯) To find, we multiply every possible outcome by the probability for that outcome and then add all these values together. 1 2 3 4 5 6 0.25 0.25 0.15 0.15 0.1 0.1 On a βfairβ die, every number has an equal chance of being rolled 1 ππ 0.166 6 This βbiasβ die is more likely to land on a 1 or 2 than a 5 or 6. 0.25 > 0.1 Find the expected value of the random variable X, where X is the number thrown. We make a table and multiply each outcome by the probability of that outcome. Playing Value Games X P(X) X.P(X) If we played a game with this die 1 0.25 0.25 where we won the β¬ amount of the number we rolled the game would be 2 0.25 0.5 worth playing if it cost less than β¬2.90. 3 0.15 0.45 4 0.15 0.6 5 0.1 0.5 6 0.1 0.6 Add all the values in X.P(X) π₯. π(π₯) 2.9 The expected value when rolling this βbiasedβ die is 2.9 It would be bad value if the game cost more than β¬2.90. For example if this game cost β¬3 to play we would lose on average β¬0.10 ever time we played.