The binomial distribution models how many successes occur in a fixed number of independent yes/no trials — like the number of heads in 10 coin flips. It applies when four conditions hold:
- A fixed number of trials (n).
- Each trial has two outcomes (success/failure).
- The probability of success (p) is the same each trial.
- Trials are independent.
The formula
The distribution’s mean is n × p and its variance is n × p × (1 − p).
Worked example
Flipping a fair coin 10 times (n = 10, p = 0.5), the expected number of heads is n × p = 5, and the formula gives the probability of any exact count (such as exactly 7 heads).
Frequently asked questions
When does the binomial distribution apply? Fixed trials, two outcomes, constant probability, independent trials.
What’s the mean of a binomial? n × p (trials times success probability).
What is “n choose k”? The number of ways to pick k successes among n trials.
As the number of trials grows large, the binomial distribution starts to look like the familiar bell-shaped normal curve. That’s why, for big samples, statisticians often approximate binomial probabilities with the normal distribution to simplify the math.