For X ~ B(n, p), the mean is np and the variance is npq, where q = 1 − p. The standard deviation is the square root of the variance.
This lesson follows calculating binomial probabilities in the probability distributions chapter.
How do I use the formulas forwards?
Identify n and p, then substitute. Let X ~ B(40, 0.3), so q = 0.7.
- Mean = np = 40 × 0.3 = 12.
- Variance = npq = 40 × 0.3 × 0.7 = 8.4.
- Standard deviation = √8.4 = 2.90 (3 s.f.).
The mean of 12 says that if you repeated the 40 trials again and again, the average number of successes would be about 12. A single group will usually land close to 12 but rarely exactly on it.
Worked example: finding n and p
A binomial variable has mean 12 and variance 7.2. Find n and p.
The key is that variance ÷ mean = npq ÷ np = q.
- q = 7.2 ÷ 12 = 0.6.
- p = 1 − 0.6 = 0.4.
- n = mean ÷ p = 12 ÷ 0.4 = 30.
Check: np = 30 × 0.4 = 12 and npq = 30 × 0.4 × 0.6 = 7.2. Both match, so n = 30 and p = 0.4.
Using the mean in a sentence
A question may say that a machine’s output is B(50, 0.08) for faulty items. The mean is 50 × 0.08 = 4, so you would expect about 4 faulty items in a batch of 50.
Use the word “expect”, and do not claim that exactly 4 will appear. The standard deviation √(50 × 0.08 × 0.92) = 1.92 tells you how far a typical batch varies from 4.
The mistake that costs marks
Writing the variance as np leaves out the factor q and makes the variance the same as the mean. That can only be right if q = 1, which means no failures at all.
A check: the variance of a binomial is always smaller than the mean, since q is less than 1. If your variance is larger than your mean, something has gone wrong.
Check yourself
A binomial variable has mean 6 and standard deviation 2. Find n and p.
Answer
Variance = 2² = 4.
q = variance ÷ mean = 4 ÷ 6 = 2/3, so p = 1/3.
n = mean ÷ p = 6 ÷ (1/3) = 18, and p = 1/3.
Check: np = 18 × 1/3 = 6 and npq = 18 × 1/3 × 2/3 = 4.
What to study next
The chapter now turns to continuous data. Continue with standardising a normal random variable. The word-problem structure worksheet helps you extract n and p from a long question.
If reverse questions slow you down, see online one-to-one Additional Mathematics tuition.