Binomial Distribution: Concepts, Formulas and Examples

The binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes, success or failure. It answers questions like: if you flip a coin 10 times, what is the probability of getting exactly 6 heads? Or: if a vaccine works 90% of the time, what is the chance that at least 8 out of 10 patients are cured?

It matters because many real situations are exactly this shape: pass or fail, defective or good. One formula replaces listing every possible outcome sequence, which makes quality control, medical trials and survey analysis practical.

Key Concepts

• Bernoulli trial: one experiment with exactly two outcomes, success (probability p) and failure (probability q = 1 - p).
• n: the number of trials, fixed before you start.
• p: the probability of success on any single trial, the same on every trial.
• X ~ Binomial(n, p): the random variable X counts the number of successes; it can take values 0, 1, 2, …, n.
• Independence: the result of one trial must not change the probability on another trial. Drawing balls from a bag without putting them back is not binomial.

Formulas

The probability mass function is P(X = k) = C(n, k) × p^k × q^(n-k), where C(n, k) = n! / (k!(n - k)!) counts the ways to arrange k successes among n trials, p^k is the probability those k trials succeed, and q^(n-k) is the probability the rest fail. The mean (expected value) is μ = np and the variance is σ² = npq, so the standard deviation is σ = √(npq). Assumptions: fixed n, independent trials, constant p, and only two outcomes per trial.

Worked Example

A die is rolled 5 times. What is the probability of getting exactly 2 sixes? Here success = six, so p = 1/6, q = 5/6, n = 5, k = 2. Step 1: C(5, 2) = 10. Step 2: p² = (1/6)² = 1/36 ≈ 0.02778. Step 3: q³ = (5/6)³ = 125/216 ≈ 0.57870. Step 4: multiply: 10 × 0.02778 × 0.57870 ≈ 0.1608. So the probability is about 0.161, or 16.1%. The mean would be np = 5/6 ≈ 0.833 sixes, and the variance npq = 5 × (1/6) × (5/6) = 25/36 ≈ 0.694.

Common Misconceptions

• "Exactly half" is not automatic: in 10 coin flips, exactly 5 heads has probability only 0.246, because the other splits share the rest.
• Trials must be independent with fixed p: drawing 5 cards from a deck without replacement is not binomial, because p changes after each draw (this is the hypergeometric distribution instead).
• The variance is not np: students forget the q in npq, which overstates the spread.

Real-World Applications

• Quality control: a factory checks 20 items and rejects the batch if more than 3 are defective, using the binomial formula to set the rule fairly.
• Medicine: a drug with an 80% cure rate is given to 15 patients; doctors compute the chance that at least 12 recover.
• Surveys and polls: estimating how likely an observed split (say 55% support in a sample of 200) could happen by chance when the true split is 50-50.

The binomial distribution turns repeated yes-or-no trials into exact probabilities: count the arrangements with C(n, k), weight by p^k q^(n-k), and remember the centre np and spread √(npq).