<\/span><\/h2>\nThe expected value of a fair game is found by multiplying each possible outcome by its respective probability and then summing them up. The formula for calculating the expected value, often denoted as E(X), is:<\/p>\n
E(X) = (Outcome 1 \u00d7 Probability 1) + (Outcome 2 \u00d7 Probability 2) + … + (Outcome n \u00d7 Probability n)<\/strong><\/p>\nLet’s walk through an example to illustrate this concept.<\/p>\n
Example:<\/strong><\/p>\nConsider a fair six-sided die. The possible outcomes are numbers 1 to 6, and each has an equal probability of 1\/6 of occurring. To find the expected value, we calculate:<\/p>\n
E(X) = (1 \u00d7 1\/6) + (2 \u00d7 1\/6) + (3 \u00d7 1\/6) + (4 \u00d7 1\/6) + (5 \u00d7 1\/6) + (6 \u00d7 1\/6) = 3.5<\/strong><\/p>\nThis means that when rolling a fair six-sided die repeatedly and taking the average of the outcomes, the expected value will converge around 3.5. If the game were fair, this would be the long-term average outcome.<\/p>\n