Unit 4: Probability, Random Variables & Probability Distributions

Binomial & Geometric Distributions

Applying the binomial and geometric probability distributions.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Mean & Variance of Random Variables

Counting Successes vs. Waiting for the First One

A player makes free throws with probability 0.75, independently each time. Before reading on: is 'how many baskets in her next 6 shots' the same kind of question as 'how many shots until her first miss' — or are these genuinely different random variables?

Definition — Binomial and Geometric Settings

A binomial setting has a fixed number of trials n, two outcomes per trial, constant probability p, and independent trials — it counts the number of successes in those n trials. A geometric setting shares the same two-outcome, constant-p, independent structure, but has no fixed number of trials — it counts how many trials occur up to and including the first success.

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Worked Example — A Binomial Probability

For n = 6 shots at p = 0.75, find P(exactly 4 makes). Conditions check: binary (make/miss), fixed n = 6, constant p = 0.75, independent shots — all satisfied. P(X=4) = C(6,4)(0.75)⁴(0.25)² = 15 × 0.31640625 × 0.0625 ≈ 0.2966, about 29.7%. The mean number of makes is μ = np = 6(0.75) = 4.5, with standard deviation √(np(1−p)) = √(6×0.75×0.25) = √1.125 ≈ 1.06 — in repeated sets of 6 shots, she typically makes about 4.5, give or take roughly one shot.

Worked Example — A Geometric Probability

Now let X count shots until her first make. Find P(first make is on the 3rd shot) — meaning the first two are misses, and the third is a make. P(X=3) = (0.25)²(0.75) = 0.0625 × 0.75 = 0.046875, about 4.7%. The mean of a geometric random variable is μ = 1/p = 1/0.75 ≈ 1.33 — since she's a good shooter, it typically takes only about 1.33 shots for her first make.

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

Ask whether the number of trials is fixed in advance (binomial — 'how many successes in exactly n tries') or determined by when success finally happens (geometric — 'how many tries until the first success') — that single question distinguishes the two settings.

Common Mistakes

Using the binomial formula for a 'trials until first success' question.
A binomial random variable requires a fixed n decided in advance — 'shots until the first make' has no fixed n, so it calls for the geometric formula instead.
Forgetting to check independence before applying either formula, e.g. when trials are drawn without replacement from a small population.
Both settings require independent trials — sampling without replacement from a small population violates this, and neither formula applies cleanly without a large-population adjustment.

Key Takeaways

A binomial random variable counts successes in a fixed number of independent trials; its mean is np.
A geometric random variable counts trials up to the first success, with no fixed number of trials; its mean is 1/p.
Both require checking the same underlying conditions — binary outcomes, constant probability, and independence — before applying either formula.

Summary

This closes Unit 4: probability rules and random variables describe chance behavior precisely. Unit 5 asks a new question — what happens to a statistic, like a sample mean or proportion, across many different random samples?

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