How should you interpret the expected value of -$0.50 per ticket in the context of a single play?
Conditions
- The expected value has been calculated as -$0.50.
- The context involves understanding the meaning of expectation in probability.
Reasoning, step by step
- Recall the definition of expected value as a probability-weighted average.
- Identify the possible outcomes for a single play: win $499 or lose $1.
- Note that neither outcome is -$0.50.
- Understand that the expected value is the center of mass of the probability distribution.
- Conclude that -$0.50 predicts the average result over a large number of trials, not the specific result of one trial.
Example
The script states: 'The expectation is dollars per ticket. It is a long-run mean, not a guaranteed result for one play.'
Common misconceptions
- Believing that a single play will result in a loss of exactly $0.50.
- Thinking that the expected value is a possible outcome of the random variable.
- Confusing the mean of the distribution with the median or mode.
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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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