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Binomial definition and conditions

Short answer

A binomial distribution gives the probability of x successes in n independent Bernoulli trials with constant success probability p, as P(X=x)=C(n,x)p^x q^(n-x). Conditions: fixed number of trials, two outcomes per trial, independence, constant p.

Define binomial distribution and explain the condition for Binomial distribution.

Definition

A random variable X that counts the number of successes in n independent Bernoulli trials, each with the same success probability p, has a binomial distribution with parameters n and p, written X ~ B(n, p):

P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, x = 0, 1, 2, …, n, q = 1 − p.

Mean = np and variance = npq.

Conditions for the binomial distribution

  1. The number of trials n is fixed in advance.
  2. Each trial has exactly two outcomes, success or failure.
  3. The trials are independent of each other.
  4. The probability of success p is constant in every trial.

Example: the number of defective bulbs in a random sample of 10 taken from a large production line, where each bulb is defective with probability 0.05.

✓ Answer checked by Claude Sonnet 5.5

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