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In probability theory the expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff). Thus, it represents the average amount one "expects" as the outcome of the random trial when identical odds are repeated many times. Note that the value itself may not be expected in the general sense - the "expected value" itself may be unlikely or even impossible. The expected value from the roll of an ordinary six-sided die is 3.5, which is not among the possible outcomes A common application of expected value is to gambling. For example, an American roulette wheel has 38 places where the ball may land, all equally likely. A winning bet on a single number pays 35-to-1, meaning that the original stake is not lost, and 35 times that amount is won, so you receive 36 times what you've bet. Considering all 38 possible outcomes, the expected value of the profit resulting from a dollar bet on a single number is the sum of what you may lose times the odds of losing and what you will win times the odds of winning The change in your financial holdings is -$1 when you lose, and $35 when you win. Thus one may expect, on average, to lose about five cents for every dollar bet, and the expected value of a one-dollar bet is $0.9473. In gambling, an event of which the expected value equals the stake (of which the bettor's expected profit is zero) is called a "fair game."
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