The binomial probability distribution answers a fundamental question: if an event has a known probability of success, what is the probability of getting exactly a certain number of successes in a fixed number of independent trials? This applies to quality control, medical testing, coin flips, and anywhere a fixed number of yes-or-no trials occur.

The Formula

The binomial probability formula calculates the probability of exactly k successes in n independent trials:

P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

Where:

  • n = number of trials
  • k = number of successes desired
  • p = probability of success in each trial
  • C(n,k) = n! / (k! × (n-k)!) — the number of combinations

C(n,k) tells you how many ways you can arrange k successes among n trials.

Worked Example

A quality inspector randomly samples 10 light bulbs from a batch known to have a 5% defect rate. What is the probability that exactly 2 bulbs are defective?

  • n = 10 trials
  • k = 2 successes (defects)
  • p = 0.05 (defect rate)
  • 1 - p = 0.95
C(10,2) = 10! / (2! × 8!) = 45
P(X = 2) = 45 × (0.05)^2 × (0.95)^8
P(X = 2) = 45 × 0.0025 × 0.6634 = 0.0746 or 7.46%

So there's a 7.46% chance of finding exactly 2 defective bulbs in that sample.

Related Probabilities

Often you want the cumulative probability — "at most 2 defects" or "at least 2 defects":

  • P(X ≤ k): Sum all probabilities from 0 to k
  • P(X ≥ k): Sum all probabilities from k to n

For large n, the binomial distribution approximates the normal distribution, which is why z-scores and normal tables are often used instead.

When to Use Binomial Probability

Use this distribution when:

  • You have a fixed number of trials
  • Each trial has two outcomes (success/failure, defective/good, yes/no)
  • The probability of success is constant
  • Trials are independent

Common applications include drug trial efficacy, election polling, manufacturing defect rates, and game outcome predictions.

Tips

The binomial formula becomes computationally heavy for large n — calculators and statistical software are essential. Also remember that this assumes independent events with a constant probability; if those assumptions break down, the result will be inaccurate.

Use our Binomial Probability Calculator to compute probabilities instantly without manual calculation.