Expected values for a game of chance

Learning Objectives Define expected value and its components for a discrete random variable. Calculate the probabilities of all outcomes in a game of chance. Construct a rational function to model the expected value of a game where a parameter (like the number of items or a payout) is variable. Solve for the conditions that make a game 'fair' by finding the roots of the rational function's numerator. Analyze the long-term behavior of a game's expected value by evaluating the limit of the rational function as the variable approaches infinity. Interpret the horizontal asymptote of an expected value function in the context of the game. Distinguish between expected payout and expected net winnings. Ever wonder if that carnival game is mathematically rigged...

TermDefinitionExample Random Variable (X)A variable whose value is a numerical outcome of a random phenomenon. It links outcomes to numbers.In a game where you roll a standard six-sided die, the random variable X could be the number that faces up. The possible values of X are {1, 2, 3, 4, 5, 6}. Probability DistributionA function or table that lists all possible values of a random variable and their corresponding probabilities. The sum of all probabilities must equal 1.For a fair die roll: P(X=1) = 1/6, P(X=2) = 1/6, ..., P(X=6) = 1/6. Expected Value (E(X))The long-run average outcome of a random experiment. It's a weighted average of the possible values of the random variable, with the probabilities serving as the weights.The expected value of a single die roll is E(X) = 1(1/6) + 2(...

Expected Value of a Discrete Random Variable E(X) = \sum_{i=1}^{n} x_i P(X=x_i) This formula calculates the expected value. You multiply each possible outcome (x_i) by its probability (P(X=x_i)) and then sum all these products together. Expected Net Winnings E(\text{Net}) = E(\text{Payout}) - \text{Cost to Play} To find the true expected outcome from a player's perspective, always subtract the cost of playing from the calculated expected payout. A positive result favors the player, a negative result favors the house. Fair Game Condition E(\text{Net}) = 0 A game is considered mathematically fair when the expected net winnings are zero. We set our expected value function equal to zero to solve for the variable value that achieves this balance.