Binomial Distribution (Discrete)¶
Models the number of successes (\(k\)) in a fixed number of independent Bernoulli trials (\(n\)).
- Notation: \(B(n, p)\)
- Function: PMF
- Formula:
\(P(X = k) = \binom{n}{k} p^{k} (1 - p)^{n - k}\)
- Mean (\(\mu\)): \(n \cdot p\)
- Variance (\(\sigma^2\)): \(n \cdot p \cdot q\)
Use Case & Example:
| Scenario | Question | \(n\) | \(p\) | \(k\) |
|---|---|---|---|---|
| Sales | What is the probability of getting exactly 3 sales (\(k=3\)) from the next 10 customers (\(n=10\)), if the per-customer purchase probability is \(0.2\) (\(p=0.2\))? | \(10\) | \(0.2\) | \(3\) |
| A/B Testing | How likely is it to see 4 out of 20 users click on a new button design, if the old click rate was \(0.1\)? | \(20\) | \(0.1\) | \(4\) |