What is the binomial distribution (plain language)
– The binomial distribution gives the probability of observing a specified number of “successes” in a fixed number of independent trials, when each trial has only two possible outcomes (commonly called success and failure) and the probability of success is the same for every trial.
– It is a discrete probability distribution: the random variable counts how many successes occur (0, 1, 2, …, n).
Key definitions
– Trial: a single repetition of the experiment (e.g., one coin flip).
– Success: the outcome of interest in a trial (e.g., heads).
– p: probability of success on each trial.
– n: number of independent trials.
– X: random variable equal to the number of successes in n trials.
– Bernoulli trial: a single two-outcome trial (a binomial with n = 1).
When to use a binomial model (checklist)
Use the binomial distribution only when all the following hold:
– Fixed number of trials (n is known in advance).
– Each trial has exactly two outcomes (success or failure).
– Probability of success p is identical for every trial.
– Trials are independent (one trial’s outcome does not affect another’s).
– You want probabilities for counts of successes (a discrete integer value).
Probability formula (the PMF)
For k = 0, 1, …, n:
P(X = k) = C(n, k) · p^k · (1 − p)^(n − k),
where C(n, k) = n! / (k!(n − k)!) is the binomial coefficient (the number of different ways to get k successes in n trials).
Mean and variance
– Mean (expected value): E[X] = n · p.
– Variance: Var(X) = n · p · (1 − p).
– Standard deviation = sqrt(n · p · (1 − p)).
Shape notes
– If p = 0.5 the distribution is symmetric around its mean.
– If p > 0.5 the distribution is skewed left (more mass at higher k).
– If p < 0.5 the distribution is skewed right (more mass at lower k).
Step-by-step: compute a binomial probability by hand
1. Identify n, p, and the target k (number of successes of interest).
2. Compute the binomial coefficient C(n, k).
3. Calculate p^k and (1 − p)^(n − k).
4. Multiply: P(X = k) = C(n, k) · p^k · (1 − p)^(n − k).
Worked numeric example 1 — coin flips (classic)
Question: What is the probability of exactly 6 heads in 20 fair coin flips?
– n = 20, k = 6, p = 0.5.
– C(20, 6) = 20! / (6!14!) = 38,760.
– p^6 · (1 − p)^(14) = (0.5)^6 · (0.5)^14 = (0.5)^20 ≈ 9.5367 × 10^−7.
– P(X = 6) = 38,760 × 9.5367 × 10^−7 ≈ 0.03696 (≈ 3.70%).
– Expected number of heads = n·p = 20 × 0.5 = 10. Variance = 20 × 0.5 × 0.5 = 5; SD ≈ 2.236.
Worked numeric example 2 — small lending illustration
Question: A small bank has 10 identical loans. Each loan has an independent 5% chance of default in one year. What is the probability exactly two loans default?
– n = 10, k = 2, p = 0.05.
– C(10, 2) = 45.
– p^
– p^2 · (1 − p)^8 = (0.05)^2 · (0.95)^8 = 0.0025 · 0.6634204313 ≈ 0.00165855.
– P(X = 2) = 45 × 0.00165855 ≈ 0.0746348 (≈ 7.46%).
– Expected number of defaults = n·p = 10 × 0.05 = 0.5.
– Variance = n·p·(1 − p) = 10 × 0.05 × 0.95 = 0.475; SD ≈ √0.475 ≈ 0.689.
Interpretation and notes:
– There is about a 7.5% chance exactly two of the ten loans default in a year under the binomial assumptions (independent loans, identical default probability).
– On average the bank expects 0.5 defaults per year; the small mean and variance indicate most of the time you’ll see 0 or 1 default.
– For small p and modest n, the Poisson approximation with λ = n·p = 0.5 gives P(X = 2) ≈ e^(−0.5)·0.5^2/2! ≈ 0.0758, which is close to the exact binomial value (useful for quick estimates).
Quick checklist to compute a binomial probability:
1. Confirm n (number of trials), k (target successes), and p (success probability per trial).
2. Compute the binomial coefficient C(n, k) = n! / (k!(n − k)!).
3. Compute p^k · (1 − p)^(n − k).
4. Multiply steps 2 and 3: P(X = k) = C(n, k) · p^k · (1 − p)^(n − k).
5. Check assumptions: trials independent, p identical across trials.
Educational disclaimer: This explanation is for educational purposes only and is not personalized investment or lending advice.
Sources
– Investopedia — Binomial Distribution: https://www.investopedia.com/terms/b/binomialdistribution.asp
– Khan Academy — Binomial random variable: https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables
– Wikipedia — Binomial distribution: https://en.wikipedia.org/wiki/Binomial_distribution
– StatTrek — Binomial Distribution Tutorial: https://stattrek.com/probability-distributions/binomial.aspx