Expected Value Example and Intuitive Explanation

The expected value is calculated as a long-run average by weighting each net outcome by its probability. First, determine the net profit for winning and losing.

Conditions: The lottery has three positions with ten choices each, allowing repetition.; The model assumes all 1000 ordered tickets are equally likely.; The ticket costs $1.; The winning prize payment is $500, which includes the full prize amount.

The net outcome for winning is $499, and the net outcome for losing is -$1. These values are derived by subtracting the $1 ticket cost from the $500 prize (for a win) and from $0 (for a loss).

Conditions: The ticket costs $1.; The winning prize is $500.; Net outcome is defined as prize minus cost.

You must distinguish them because expected value measures the average financial gain or loss relative to the initial investment. The prize paid is the gross revenue, but the net profit accounts for the cost of the ticket.

Conditions: The lottery ticket has a non-zero cost.; The calculation aims to determine the player's expected financial outcome (profit/loss).

The expected value of -$0.50 should be interpreted as a long-run average loss per ticket, not as a guaranteed result for a single play. In any individual game, the player will either win $499 or lose $1; the -$0.50 figure represents the mean outcome if the game were played many times.

Conditions: The expected value has been calculated as -$0.50.; The context involves understanding the meaning of expectation in probability.

The probability of winning is $1/1000$. This is derived from the fact that there are 1000 equally likely ordered tickets, and only one of them is the winning ticket.

Conditions: The lottery consists of three positions.; Each position has ten possible choices (digits 0-9).; Repetition of digits is allowed.; The model assumes a uniform distribution over all possible tickets.