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Independence and combinatorics

Two events are independent when $P(A\cap B) = P(A)P(B)$ — one happening does not change the probability of the other. Factorial $n!$ counts the number of orderings of $n$ distinct things, while the binomial coefficient $\binom{n}{k}$ counts the number of ways to choose $k$ of $n$ when order does not matter.

P(A\cap B) = P(A)P(B)

requirement for independence

\binom{n}{k} = \frac{n!}{k!(n-k)!}

number of ways to choose

k

n

Symbols

$n$	total number of items
$k$	number chosen

Example

Choosing 3 of 4 components: $\binom{4}{3} = 4$ ways.

Ordering 6 distinct parts: $6! = 720$ orderings.

Disjoint events (cannot happen together) are almost never independent — they are in fact strongly dependent.

Practise probability for free →

← Bayes' theorem · Normal distribution and z-score →

Part of Statistics and Risk Analysis: Probability.