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Expected Value
Long-Run Average of a Random Outcome
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Expected value is a way to describe the average outcome you would expect from a random process over many repeated trials. It combines each possible outcome with how likely that outcome is, so it is more informative than just listing outcomes alone. Expected value is used in games, insurance, finance, quality control, and scientific modeling. It helps people compare uncertain choices using one summary number.
The key idea is to multiply each outcome by its probability and then add the results. This gives a probability weighted average, which may or may not be one of the actual outcomes you can get in a single trial. For a discrete random variable X, the expected value is E(X) = Σ[x·P(x)]. In the long run, as the number of trials becomes large, the sample average tends to get closer to the expected value.
Key Facts
- Expected value for a discrete random variable: E(X) = Σ[x·P(x)]
- Probabilities must add to 1: ΣP(x) = 1
- Expected value is a long run average, not a guaranteed single result
- For a fair six sided die: E(X) = (1+2+3+4+5+6)/6 = 3.5
- Linearity of expectation:
- For a game payoff, expected net value = Σ[payoff·probability] - cost
Vocabulary
- Expected value
- The probability weighted average of all possible values of a random variable.
- Random variable
- A variable that assigns a numerical value to each outcome of a random process.
- Probability
- A number from 0 to 1 that tells how likely an outcome is.
- Discrete distribution
- A list or rule giving the probabilities of distinct separate outcomes.
- Long run average
- The average result approached when a random experiment is repeated many times.
Common Mistakes to Avoid
- Adding outcomes without weighting by probability, which is wrong because expected value depends on both the size of each outcome and how likely it is.
- Using probabilities that do not add to 1, which is wrong because a valid probability distribution must account for all possible outcomes completely.
- Thinking expected value must be one of the actual outcomes, which is wrong because it is an average and can fall between possible results such as 3.5 for a die roll.
- Confusing expected value with the most likely outcome, which is wrong because the highest probability outcome and the probability weighted average can be different numbers.
Practice Questions
- 1 A game pays 4 with probability 0.5, and loses $6 with probability 0.2. Find the expected value of one play.
- 2 A random variable X takes values 2, 5, and 9 with probabilities 0.25, 0.5, and 0.25. Calculate E(X).
- 3 A lottery ticket has a positive expected prize before the ticket cost is subtracted, but a negative expected net value after cost. Explain what this means about playing the lottery many times.