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What is Binomial distribution?

The probability distribution of the number of successes when a trial with success probability p is repeated n times independently. The probability of exactly k successes is nCk × pᵏ × (1 − p)ⁿ⁻ᵏ, the mean is np and the variance is np(1 − p). For example, when 10 parts are taken from a population with a 10% defect rate, the probability that none is defective is 0.9¹⁰ ≈ 0.349. It is used to calculate the acceptance probability of sampling inspection and the number of bit errors in communication, and when n is large and p is small it can be approximated by a Poisson distribution.

Japanese name: 二項分布

Binomial distribution appears in 2 practice questions across 2 topics, including Reliability, Probability, Statistics and Queueing.

Field: Probability, Statistics and Queueing

Try it: questions about Binomial distribution

Which distribution, often used in reliability data analysis, can represent early-failure, random-failure and wear-out types depending on its shape parameter?

  1. AWeibull distribution
  2. BUniform distribution
  3. CBinomial distribution
Show answer and explanation

Answer: A. Weibull distribution

In the Weibull distribution, a shape parameter m below 1 describes infant mortality with decreasing failure rate, m = 1 a constant random-failure rate, and m above 1 wear-out with increasing rate. Its strength is that the same family of distributions covers each of the three bathtub periods just by changing m. The failure rate of a single Weibull distribution only rises or falls monotonically, however, so the whole bathtub curve is modeled by combining distributions with different m. Failure data plotted on Weibull probability paper form a straight line whose slope is m. A uniform distribution gives equally likely values, and a binomial distribution counts successes.

Five products are taken from a process with a defect rate of 20%. Whether each product is defective is independent of the others. Which of the following is closest to the probability that exactly one of the five is defective?

  1. A0.082
  2. B0.33
  3. C0.41
Show answer and explanation

Answer: C. 0.41

The number of defectives follows a binomial distribution with n = 5 and p = 0.2. The probability of exactly one defective is the 5C1 = 5 ways of choosing which of the five is defective multiplied by the probability 0.2 × 0.8⁴ = 0.08192 that this one is defective and the other four are good: 5 × 0.08192 = 0.4096 ≈ 0.41.

Binomial distribution in the explanations

Questions that mention Binomial distribution (2)

  1. Which distribution, often used in reliability data analysis, can represent early-failure, random-failure and wear-out types depending on its shape parameter?
  2. Five products are taken from a process with a defect rate of 20%. Whether each product is defective is independent of the others. Which of the following is closest to the probability that exactly one of the five is defective?

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Related terms

Weibull distribution Bathtub curve Bayes' theorem Conditional probability Confidence interval Correlation coefficient Expected value Exponential distribution Hypothesis testing Median

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