Binomial Theorem
-- ①
-- ②
If we subtract ② from ①, we get
Number of Outcomes | With Replacement | Without Replacements |
---|---|---|
2 | Binomial (different when ...) | Hypergeometric |
3 | Multinomial | Multivariate Hypergeometric |
- success → geometric
- success → negative binomial
In Probability,
Proof
Base case
Let . Then
Induction Hypothesis
Assume that . Let . Then
Let and for the first and for the second sum. Then,