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Probability Unit 5: Binomial, Geometric and Poisson Distributions

A few named discrete distributions model most counting situations: successes in fixed trials, waiting times, and rare events. Knowing their PMFs, means and variances, and when each applies, is essential for every probability exam.

Covers: Bernoulli trials and the binomial, Geometric distribution and memorylessness, Negative binomial, Hypergeometric distribution, Poisson distribution, Poisson approximation to the binomial, Sums and thinning of Poisson counts

Study guide

A Bernoulli trial has two outcomes, success with probability p and failure with probability 1 − p. The binomial(n, p) distribution counts successes in n independent Bernoulli trials with the same p, with PMF C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ, mean np and variance np(1 − p). Check the conditions before using it: a fixed number of trials, independence and a constant p. Drawing cards or sampling parts without replacement violates independence; the exact count there is hypergeometric, P(X = k) = C(K, k)C(N − K, n − k)/C(N, n), with the same mean nK/N but a smaller variance. When the sample is a small fraction of the population, the binomial is a good approximation.

The geometric distribution models the trial on which the first success occurs: P(X = k) = (1 − p)ᵏ⁻¹p for k = 1, 2, …, with mean 1/p and variance (1 − p)/p². Some books instead count the failures before the first success, which shifts the mean to (1 − p)/p, so always check the convention. The geometric is memoryless: P(X > m + n | X > m) = P(X > n), so a run of failures does not make a success 'due'. The negative binomial generalizes it to the trial of the r-th success, with P(X = k) = C(k − 1, r − 1)pʳ(1 − p)ᵏ⁻ʳ and mean r/p.

The Poisson(λ) distribution, P(X = k) = e^(−λ)λᵏ/k!, counts events that occur independently at a constant average rate, such as calls per hour or claims per year. Its mean and variance both equal λ, and λ scales with the length of the interval: a rate of 3 per hour gives λ = 1.5 for 30 minutes. The Poisson also approximates the binomial when n is large and p is small, with λ = np. Its most likely value is ⌊λ⌋ when λ is not an integer.

Poisson counts combine neatly. The sum of independent Poisson variables is Poisson with the means added. If each event is independently kept with probability p (thinning), the kept events are Poisson(λp). Conversely, given X + Y = n for independent Poissons, X is binomial(n, λ₁/(λ₁ + λ₂)). Sports series and 'r-th success' questions are negative binomial problems in disguise. On exams, the main skill is choosing the right model from the wording, then using the complement for 'at least one' and summing a few terms for 'at most k'.

Key ideas

• Binomial: fixed n independent trials with constant p.
• Without replacement from a finite population, the count is hypergeometric.
• Geometric: waiting time to the first success, mean 1/p, memoryless.
• Negative binomial: waiting time to the r-th success, mean r/p.
• Poisson: counts of events at a constant rate, mean = variance = λ.
• Poisson(np) approximates binomial(n, p) for large n and small p.
• Independent Poissons add, and thinning a Poisson gives a Poisson.

Formulas and rules

Common mistakes

• Using the binomial for sampling without replacement from a small population.
• Forgetting to rescale the Poisson mean when the time interval changes.
• Mixing up the 'trials' and 'failures' versions of the geometric distribution.
• Omitting the C(n, k) factor and computing the probability of one specific order.
• Believing a success becomes more likely after a run of failures, which memorylessness rules out.

Before the test, make sure you can

• Identify which named distribution fits a word problem.
• Calculate binomial probabilities, means and variances, including 'at least one'.
• Calculate geometric probabilities and use memorylessness.
• Calculate negative binomial probabilities for the r-th success.
• Compute Poisson probabilities with a rate scaled to the interval.
• Use the Poisson approximation to the binomial.
• Combine independent Poisson counts and apply thinning.
• Distinguish binomial from hypergeometric sampling.

Words to know

Bernoulli trial

A single trial with two outcomes, success (probability p) and failure.

Binomial distribution

Number of successes in n independent trials with constant p; PMF C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ.

Binomial mean and variance

np and np(1 − p).

Geometric distribution

Trial number of the first success; P(X = k) = (1 − p)ᵏ⁻¹p, mean 1/p.

Memoryless property

P(X > m + n | X > m) = P(X > n); held by the geometric and exponential distributions.

Negative binomial distribution

Trial number of the r-th success; mean r/p.

Hypergeometric distribution

Number of successes in a sample of n drawn without replacement from N items containing K successes.

Poisson distribution

Count of events at constant rate; P(X = k) = e^(−λ)λᵏ/k!.

Poisson mean and variance

Both equal λ.

Poisson approximation

Binomial(n, p) ≈ Poisson(np) for large n and small p.

Poisson process

Events occurring independently at a constant rate, with Poisson counts in any interval.

Poisson thinning

Keeping each event independently with probability p turns Poisson(λ) into Poisson(λp).

Sum of Poissons

Independent Poisson(λ₁) and Poisson(λ₂) add to Poisson(λ₁ + λ₂).

Finite population correction

The factor (N − n)/(N − 1) by which the hypergeometric variance is smaller than the binomial variance.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.