How do you scale an expected value from a single unit to multiple independent units?
Conditions
- The expectation for a single unit is known or calculable.
- The number of units is fixed and finite.
Reasoning, step by step
- Determine the expected value for one unit by summing the products of outcomes and their probabilities.
- Identify the total number of units involved.
- Multiply the single-unit expected value by the number of units.
- Interpret the result as the total expected value for the aggregate.
Example
If the expected profit per laptop is 23.8, the expected profit for 100 laptops is .
Common misconceptions
- Thinking that you must simulate all possible combinations of outcomes for multiple units.
- Believing that scaling only works if the units are statistically independent (linearity of expectation applies generally).
- Confusing the scaled expected value with the most likely total outcome.
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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.
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).
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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