How do you compute the expected value of a three-digit lottery ticket with a $500 prize and a $1 cost under a uniform model?
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.
Reasoning, step by step
- Calculate the total number of possible tickets: .
- Determine the probability of winning: .
- Determine the probability of losing: .
- Calculate the net outcome for a win: dollars.
- Calculate the net outcome for a loss: dollar.
- Compute the expected value: .
- Simplify the expression: .
- Interpret the result as the long-run mean profit per ticket.
Example
The script states: 'If the 500-dollar payment includes the full prize, net outcomes are 499 on a win and −1 on a loss. The expectation is dollars per ticket.'
Common misconceptions
- Confusing the gross prize ($500) with the net profit ($499).
- Believing the expected value guarantees a loss of exactly $0.50 on a single play.
- Assuming the probability of winning is different if the digits are not independent or if repetition is not allowed.
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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 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).
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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