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Binomial Distribution

➗ Mathematics & Statistics By CampusDown Wiki Editorial Team Last updated
Quick answer: the binomial distribution gives the probability of each possible number of successes in n independent trials with the same success probability.
Contents
  1. 1. Conditions and formula
  2. 2. Mean, variance and an example
  3. 3. Approximations and uses
  4. 4. References

Conditions and formula

A trial with only two outcomes is a Bernoulli trial. If (1) the number of trials n is fixed, (2) trials are independent and (3) the success probability p is constant, the number of successes X follows B(n, p).

P(X=k) = C(n,k) · pk · (1-p)n-k, for k = 0, 1, …, n

Mean, variance and an example

  • Mean np, variance np(1-p), standard deviation √(np(1-p)).
  • Example: if a surgery succeeds 80% of the time, the probability that exactly 10 of 15 patients succeed is C(15,10)·0.810·0.25 ≈ 0.103; the expected number is 12.

Approximations and uses

When n is large and both np and n(1-p) are at least about 5 to 10, the normal distribution N(np, np(1-p)) approximates it. When n is large and p small, the Poisson distribution with λ = np does. It is used for defect counts in quality control, poll proportions, treatment success counts and tests of proportions.

References

  1. Ross, S. M. (2019). A First Course in Probability (10th ed.). Pearson.
  2. Hogg, R. V., Tanis, E. A., & Zimmerman, D. L. (2019). Probability and Statistical Inference (10th ed.). Pearson.
  3. Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley.

Reports and materials that use this theory (Korean)

There are no materials dedicated to this theory yet, but you can search related reports in the CampusDown library (Korean).

Find more reports · Search '이항분포' on CampusDown →
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This article was last updated on October 15, 2026. It is based on widely recognized original works and textbooks; when citing it in a paper, please check the original sources listed in the references.