Definition: A binomial distribution gives the probabilities of the number of successes in a fixed number of independent trials, each with the same probability of success.
When is a situation binomial?
Four conditions: a fixed number of trials n, two outcomes (success or failure), independent trials, and the same success probability p each time. The count of successes X then follows a binomial distribution.
What are the formulas?
- P(X = k) = C(n, k) pk (1 – p)n – k
- Mean = np
- Standard deviation = square root of np(1 – p)
Worked example: a factory part is defective with probability 0.3, and 10 parts are tested. The chance of exactly 3 defects is C(10, 3)(0.3)3(0.7)7 = 120 times 0.027 times 0.0824, about 0.267.
The expected number of defects is 10(0.3) = 3, and the standard deviation is the square root of 2.1, about 1.45.
What goes wrong?
Forgetting the combination factor C(n, k), which counts the orders of successes. Mathbot also checks independence: drawing without replacement from a small group breaks it.