Binomial distribution
The binomial distribution counts successes in n independent trials, each with the same success chance p. P(X = k) = nCk x p^k x (1-p)^(n-k). The mean is np and the variance is np(1-p).
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Analogy
Shooting five free throws with a fixed 60 percent chance each, and counting how many go in.
Example
n = 5, p = 0.6P(X=3) = 10 x 0.6^3 x 0.4^2= 10 x 0.216 x 0.16 = 0.3456Mean = 5 x 0.6 = 3Variance = 5 x 0.6 x 0.4 = 1.2
In practice
Used for yes/no counts: defective items in a batch, voters who say yes, patients who recover.
Practice
Flashcards
- Mean and variance of a binomial?Mean np, variance np(1-p).
Interview questions
- A shooter scores 60% of free throws. Find the chance of exactly 3 scores in 5 shots, and state the binomial conditions you assumed.
- n = 5, p = 0.6
- P = 10 x 0.216 x 0.16 = 0.3456
- Fixed number of trials
- Independent trials, same p
Questions people ask
What conditions must hold for a binomial model?
There is a fixed number of trials n. Each trial has two outcomes, success or failure. The success chance p is the same every time, and trials are independent. Drawing cards without replacement breaks independence, though the model works well for a small sample from a huge population.
Where does the nCk come from in the binomial formula?
It counts the different orders in which k successes can occur among n trials. Each particular order has probability p^k x (1-p)^(n-k). Multiplying by the number of orders gives the total. With 3 successes in 5 trials there are 10 orders.