Why is expected profit from a probability-weighted average not a guaranteed realized profit?
Conditions
- The scenario involves random outcomes with assigned probabilities.
- The calculation uses the probability-weighted average (expectation).
Reasoning, step by step
- Calculate the expected value per unit using the weighted mean of gains and losses.
- Multiply by the number of units to get the total expected profit.
- Recognize that this total is an average over many hypothetical repetitions.
- Understand that individual realizations will deviate from this average due to randomness.
Example
In the laptop example, the mean is 23.8 per unit. Summing expectations gives 2380 for 100 units, but this is not a guaranteed realized profit because actual sales outcomes will vary.
Common misconceptions
- Believing that the expected value is the exact amount you will receive in a single trial.
- Assuming that averaging over a small number of trials eliminates variance completely.
- Confusing the mathematical expectation with a deterministic prediction.
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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).
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.
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