Why must you distinguish the prize paid from the net profit when calculating the expected value of a lottery ticket?
Conditions
- The lottery ticket has a non-zero cost.
- The calculation aims to determine the player's expected financial outcome (profit/loss).
Reasoning, step by step
- Identify the cost of the ticket (e.g., $1).
- Identify the gross prize paid upon winning (e.g., $500).
- Calculate the net profit for a win by subtracting the cost from the prize ($500 - $1 = $499).
- Calculate the net profit for a loss by subtracting the cost from zero ($0 - $1 = -$1).
- Use these net values in the expected value formula.
- Recognize that using the gross prize ($500) instead of net profit ($499) ignores the sunk cost of the ticket.
Example
The script states: 'When a ticket costs one dollar, distinguish the prize paid from the net profit after the ticket cost.' It later clarifies: 'If the 500-dollar payment includes the full prize, net outcomes are 499 on a win and −1 on a loss.'
Common misconceptions
- Thinking that the expected value is simply the probability of winning multiplied by the prize amount.
- Believing that the ticket cost is irrelevant to the expected value calculation if the prize is large.
- Confusing the concept of 'revenue' with 'profit' in probability calculations.
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Related questions
To calculate the expected value, list all mutually exclusive outcomes, assign their stated monetary gains or losses, and ensure the probabilities sum to one. Then, compute the weighted mean by multiplying each outcome's value by its probability and summing these products.
Conditions: The outcomes are mutually exclusive.; The probabilities assigned to the outcomes sum to exactly one.; Monetary values are defined consistently, including any applicable costs.
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.
To scale an expected value, calculate the expectation for a single unit and then multiply it by the total number of units. This relies on the linearity of expectation, which holds regardless of whether the units are independent, provided the individual expectations are defined.
Conditions: The expectation for a single unit is known or calculable.; The number of units is fixed and finite.
Expected profit is a long-run average calculated by weighting outcomes by their probabilities. It does not guarantee that any single trial or even a finite series of trials will yield exactly that amount, because actual results vary randomly around the mean.
Conditions: The scenario involves random outcomes with assigned probabilities.; The calculation uses the probability-weighted average (expectation).
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