Discrete Probability Distributions
Learning Objectives
- Understand the properties of a discrete probability density function
- Find the mean of a discrete random variable from its probability density function
- Find the standard deviation of a discrete random variable from its probability density function
Why This Matters
Every insurance premium you pay was set by two numbers pulled from a probability distribution. The mean claim amount tells the insurer what to expect to pay per policy on average, but the standard deviation tells them how wildly actual claims can swing from that average -- a hurricane year versus a quiet year. Without the mean, they can't price the policy; without the standard deviation, they can't survive the bad years. The same two computations drive investment risk models, quality control in manufacturing, and the expected-value calculations behind every casino game you'll ever play.
How to Use This Simulation
- Select a preset distribution or build your own in Custom mode, then adjust probabilities using the sliders and watch the bar chart, mean line, and computations update in real time.
- Watch the validity indicator -- probabilities must sum to exactly 1.0 before the mean and standard deviation can be computed.
- Switch to the Step-by-Step Computation tab to see every term in the mean, variance, and standard deviation formulas with substituted values.
- Check the Explanation Panel below -- it updates as you interact and tells you what the numbers mean.
Computation Table
| xi | P(xi) | xi · P(xi) | (xi − μ)² | (xi − μ)² · P(xi) |
|---|
What's Happening
Quick Check
A fair six-sided die has the probability distribution: each outcome (1, 2, 3, 4, 5, 6) has probability 1/6. A student computes the mean as μ = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5 and says, "The expected value is 3.5, so I should expect to roll a 3.5." Which statement best describes the error in the student's interpretation?
Try This
A local bakery tracks how many items are returned each day. The probability distribution is: 0 returns (P = 0.15), 1 return (P = 0.30), 2 returns (P = 0.25), 3 returns (P = 0.20), 4 returns (P = 0.10). Compute the mean number of daily returns by hand using μ = Σ xi × P(xi). Enter the distribution into the simulation using Custom mode and verify your answer. What does the mean tell the bakery manager about a "typical" day?
Using the same bakery distribution from the Starter: (1) Verify the distribution is valid -- confirm ΣP(x) = 1 and each probability is between 0 and 1. (2) Compute the mean μ. (3) Compute the standard deviation: for each outcome, calculate (xi − μ)², multiply by P(xi), sum all terms, then take the square root. Verify each step against the simulation's Step-by-Step Computation tab. In one sentence, explain why computing the standard deviation required you to know the mean first.
Two food delivery apps both report an "average delivery time" of 30 minutes for a specific restaurant, but their probability distributions differ:
App A: 25 min (0.10), 28 min (0.20), 30 min (0.40), 32 min (0.20), 35 min (0.10)
App B: 20 min (0.30), 25 min (0.15), 30 min (0.10), 35 min (0.15), 40 min (0.30)
(1) Verify both distributions are valid. (2) Compute the mean for each -- confirm they're both 30 minutes. (3) Compute the standard deviation for each. (4) You're ordering food for a meeting that starts in exactly 35 minutes. Which app do you choose? Defend your answer in two sentences using the standard deviations you calculated.
Instructor Notes
Teaching Notes
This simulation works best when you start with the Fair Die Roll preset and ask students to predict where the mean line will appear before they look. Most students predict 3 (the middle outcome) instead of 3.5 (the weighted average). The surprise that the mean falls between outcomes is the entry point for the "expected value is a weighted average" insight.
The "same mean, different spread" comparison is built into the presets. Load Daily Support Tickets (μ = 1.5500), then switch to Goals per Soccer Match (μ = 1.5500). Ask students which scenario is more predictable. The standard deviations (1.3739 vs 1.1170) quantify the intuitive difference they'll see in the bar chart shapes.
Common Student Errors
- Computing the mean as a simple average of outcomes (e.g., (1+2+3+4+5+6)/6 = 3.5 "by accident") instead of the weighted average. This gives the correct answer only for uniform distributions, so the error is invisible on the Fair Die Roll preset. Switch to a non-uniform preset to expose it.
- Forgetting to square the deviations before weighting by probability when computing variance. The simulation's Step-by-Step tab shows every intermediate value.
- Confusing variance and standard deviation. Students sometimes report σ² when asked for σ. Emphasize that σ is in the original units of the outcome.
- Believing the mean must be one of the possible outcomes. The Fair Die Roll's μ = 3.5 directly confronts this.
Discussion Questions
- A casino game pays $0 with probability 0.60, $5 with probability 0.30, and $20 with probability 0.10. The game costs $4 to play. Is this a good deal? How does the expected value help you decide?
- Two investments both have an expected return of 8% per year. One has σ = 2% and the other has σ = 15%. A retiree and a 25-year-old are both investing. Should they choose the same option? Why or why not?
- If you add 10 to every outcome in a distribution, what happens to the mean? What happens to the standard deviation? Why?
Exam Connection
Typical exam questions provide a probability distribution table and ask students to (a) verify it is valid, (b) compute the mean, and (c) compute the standard deviation. The Step-by-Step Computation tab mirrors this exact workflow, with every intermediate calculation visible. Some exam items ask students to interpret the mean and SD in context -- the Challenge tier practices this judgment directly.