Technique 1: Direct Counting For equally likely outcomes, P = . Use combinations for unordered selection, permutations for ordered.
Technique 2: Complement Counting P(at least one) = 1 - P(none). Most efficient when "at least one" requires many cases but "none" is a single case.
Technique 3: Sequential Counting (Multiplication Rule) For multi-stage experiments: P(A and B) = P(A)*P(B|A). Used for drawing without replacement.
Technique 4: Partitioning Split into mutually exclusive cases and add. P(A) = P(A and B) + P(A and B').
Common counting setups:
| Setup | Formula | Example |
|---|---|---|
| n dice thrown | 6^n total outcomes | P(all different) |
| n coins tossed | 2^n total outcomes | P(k heads) = C^n |
| r from n without replacement | C(n,r) | Drawing balls from bag |
| r from n with replacement | Rolling dice | |
| Derangements | = n!(1-1+1/2!-...) | No letter in correct envelope |
Key identities:
- C(n,r) = C(n, n-r) — choosing r to include = choosing n-r to exclude
- C(n,0)+C(n,1)+...+C(n,n) = 2^n — total subsets
- C(n,r) = C(n-1,r-1)+C(n-1,r) — Pascal's identity