Lesson 2: 6.2: Analyzing Discrete Random Variables

If you play a game of chance 1,000 times, why is the average result more predictable than the result of a single game?

What does the "Expected Value" tell us about the fairness of a bet or an insurance policy?

How does the standard deviation help us measure the "risk" of a random variable?

Mean of a random variable

Expected Value

Variance

Standard deviation

HSS-MD.A.2: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

HSS-MD.B.5: Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

The SAT often includes "Weighted Average" problems. Expected value is the ultimate weighted average where the "weights" are the probabilities. Students may see tables of data and be asked for the "average" value, which requires this exact logic.

This section moves from describing a distribution to summarizing it with numbers. It’s about the "long-run" behavior of random phenomena.

DOK 1: State the formula used to calculate the mean of a discrete random variable.

DOK 2: Calculate the expected profit for a small business based on a probability distribution of daily sales.

DOK 3: Given two different "Side Hustle" opportunities with the same expected value but different standard deviations, justify which one a "risk-averse" person should choose.

The Problem: A life insurance company sells a $250,000 one-year term policy to a 20-year-old female for $200. According to vital statistics, the probability that a female of this age survives the year is 0.99944.

Task 1: Create a probability distribution for the insurance company’s profit.

Task 2: Calculate and interpret the expected value.

Students often think the expected value must be a possible outcome. If the EV of a die roll is 3.5, they might think they did the math wrong because you can't roll a 3.5. Remind them it is a long-run average, not a "prediction."

Standard Deviation vs. Variance: Students often forget to take the square root at the end of the variance formula.

Support: Use the TI-84

Extension: Introduce the "House Edge." Have students research the expected value of a bet in Roulette or Craps. Ask them to explain why casinos are guaranteed to make money even if someone wins a "jackpot."

Teacher assigns examples from the textbook and other resources.

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