Final answer:
The expected value of a game is found by multiplying each payoff by its probability and summing the results. It represents the long-term average gain or loss per game, allowing one to understand what they might expect to win or lose over many plays of the game.
Step-by-step explanation:
Understanding Expected Value
The expected value of a game is calculated by multiplying each potential payout (ai) by its corresponding probability (pi) and summing these products together. Formally, it can be written as:
E[X] = a1p1 + a2p2 + ... + anpn
This expected value represents the average amount one can expect to win or lose per game if the game were played a very large number of times. It is a measure of the central tendency or long-term average of the game's payouts taking into account the probabilities of each outcome.
The significance of expected value lies in its ability to provide a single number summarizing the average outcome of a random event given its probabilities and associated payoffs. For example, if a game has a positive expected value, it suggests that over time, one could expect to make money playing this game. Conversely, a negative expected value indicates that on average, one would lose money.