Binomial distribution: at most k
"At most k" means adding every value from 0 up to k.
Worked example
5 parts are checked; each is faulty with probability 0.30. What is P(at most 3 faulty)?
Worked solution
- 1.X ~ B(5; 0.30)
- 2.P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
- 3.= 0.1681 + 0.3601 + 0.3087 + 0.1323 = 0.9692
Typical mistake: Leaving out P(X = 0) for "at most k".
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task 1 of 3Binomial distribution
Task
10 parts are checked; each is faulty with probability 0.20. What is P(at most 3 faulty)?
P(X ≤ 3) (4 decimal places)
Comma or point; fractions like 1/6 work too.
Formulas for this topic
- P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
- P(X ≤ k) = P(X = 0) + … + P(X = k)
- E(X) = n · p, σ = √(n · p · (1 − p))
Typical mistake: Leaving out P(X = 0) for "at most k".
Frequently asked questions
What is the typical mistake in "Binomial distribution: at most k"?
Leaving out P(X = 0) for "at most k".
How do I work out "at least k"?
Use the complement: P(X ≥ k) = 1 − P(X ≤ k − 1).