How do you calculate the expected value from an explicit outcome table with mutually exclusive outcomes?
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.
Reasoning, step by step
- List all mutually exclusive outcomes.
- Assign the stated gain or loss to each outcome.
- Verify that the probabilities of all outcomes sum to one.
- Multiply each outcome's monetary value by its corresponding probability.
- Sum the resulting products to find the expected value.
Example
For outcome values 500 and −100 with probabilities 0.2 and 0.8, the weighted mean is calculated as .
Common misconceptions
- Forgetting to check if the probabilities sum to one before calculating.
- Including costs inconsistently across different outcomes.
- Treating non-mutually exclusive events as separate rows without adjusting probabilities.
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Related questions
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).
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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