Probability

Discrete Probability Distributions

Constructing and interpreting a probability distribution table.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Independence & Conditional Probability

The Long-Run Average of a Chance Outcome

A survey of a class's pet ownership finds: 30% of students own 0 pets, 35% own 1, 20% own 2, and 15% own 3. Before reading on: since no student actually owns a fractional number of pets, what would it even mean to compute one 'average' number of pets for the whole class?

Definition — Discrete Probability Distribution

A discrete random variable's probability distribution lists every possible value alongside its probability, with the probabilities summing to 1. Its mean (expected value), E(X) = Σx·p(x), is a probability-weighted average — the long-run average outcome, not a prediction about any single individual. Its standard deviation, from Var(X) = Σ(x−μ)²p(x), measures typical spread around that mean.

E(X) = \sum x \cdot p(x)

Worked Example — Finding the Mean and Standard Deviation of a Discrete Random Variable

Checking the probabilities sum to 1: 0.30 + 0.35 + 0.20 + 0.15 = 1.00. Mean: E(X) = 0(0.30) + 1(0.35) + 2(0.20) + 3(0.15) = 0 + 0.35 + 0.4 + 0.45 = 1.20 pets. This doesn't mean any single student owns 1.2 pets — every actual student owns 0, 1, 2, or 3 — it means the average across the whole class, in the long run, works out to 1.2. Variance: (0−1.2)²(0.30) + (1−1.2)²(0.35) + (2−1.2)²(0.20) + (3−1.2)²(0.15) = 0.432 + 0.014 + 0.128 + 0.486 = 1.06, so the standard deviation is √1.06 ≈ 1.03 pets — a typical student's pet count differs from the class average by about 1 pet.

Probability Simulator

Experiment

Type

Sides

Number of dice

Run a trial to see the outcome here.

Simulation

Target trials

Trials per tick

Track outcome

0 / 200 trials

Histogram — experimental vs theoretical

Running probability — Law of Large Numbers

Statistics

Mean

— experimental

3.5 theoretical

Variance

— experimental

2.917 theoretical

Standard deviation

— experimental

1.708 theoretical

Trials

0 experimental

— theoretical

Frequency table

Outcome	Experimental	Theoretical
1	—	0.167
2	—	0.167
3	—	0.167
4	—	0.167
5	—	0.167
6	—	0.167

Central Limit Theorem

Draws 300 independent samples of 5trials each from the current experiment, averages each sample, and histograms the resulting sample means — the distribution should look progressively more bell-shaped (normal) as the sample size grows, even though the experiment’s own distribution usually isn’t.

Sample size

Number of samples

Tip

The expected value doesn't need to be (and usually isn't) one of the random variable's actual possible values — it's a weighted average across all of them, exactly like 1.2 pets here.

Common Mistakes

Treating the expected value as the single most likely outcome.
The most likely individual outcome here is 1 pet (probability 0.35, the largest single probability) — E(X) = 1.2 is the probability-weighted average across all four outcomes, a different quantity from the mode.

Key Takeaways

A discrete probability distribution lists every possible value of a random variable with its probability, summing to 1.
E(X) = Σx·p(x) is a long-run average across many repetitions, not a prediction about any single outcome.
The variance and standard deviation of a random variable measure typical spread around that long-run average, the same way they do for a sample.

Summary

This closes the unit on probability. The final unit uses these ideas to reason about how a sample statistic behaves across many possible samples — the foundation for estimating an unknown population value.

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