Discrete random variables – expectation as the long-run average
MathematicsProbabilityAges 17–18
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Probability
Build the probability distribution table of a discrete random variable in a real-world scenario (a dice game with a prize, the number of correct guesses on a multiple-choice quiz, the number of defective items in a batch), then compute the expectation, variance and standard deviation, and run thousands of simulated trials to watch the experimental mean converge on the expectation — and so decide whether the game favours the player.
Lesson: Discrete random variables and their key measures
What it shows
A discrete random variable X takes values xᵢ with probabilities pᵢ that add up to 1. Its expectation E(X) = Σxᵢpᵢ is the long-run average, the variance V(X) measures spread and σ = √V is the standard deviation. The quiz and defective-item scenarios follow the binomial distribution B(n, p), where E = np and V = np(1 − p). In the dice game each play costs 10 and E(X) = (prize − 50)/6, so the game is fair only when the prize for a 6 is 50. Simulated trials show the experimental mean settling towards E(X), the law of large numbers.
How to use
Choose a Scenario, set n and p (and the prize in the dice game), and read the distribution table with E(X), V(X) and σ. Ask students to decide from the expectation whether the game is worth playing before any simulation. Then press +1 trial a few times, followed by +100 and +1000, and watch the red line of the running average swing at first and then settle near E(X). Press Clear to repeat the run and see a different path.
Parameters you can change
- Scenario Dice game with a prize, Guessing on a multiple-choice quiz, Defective items in a batch
- Number of plays / questions / batch size 1–40
- Success probability / defect rate 0.01–0.99
- Prize for rolling a 6 (game) 0–100
- Currency From the language, US dollar, Euro, Indian rupee, Indonesian rupiah, Saudi riyal
Questions to explore
- Guessing 10 questions with 4 options each, how many correct answers do you expect on average?
- What is the smallest prize for a 6 that makes the dice game favour the player?
- What does a large variance tell you about the risk of a single play or batch?
