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Combinatorics and Basic Probability

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Combinatorics and Basic Probability

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Now · 10.1 Counting Principles

Combinatorics and Basic Probability

10.1 Counting Principles

Product rule: If task A has

m

outcomes and task B has

n

, then the sequence (A then B) has

m \times n

outcomes. Sum rule: If tasks A and B are mutually exclusive, then A or B has

m + n

outcomes.

10.2 Permutations and Combinations

Permutation (order matters):

P(n, k) = \frac{n!}{(n-k)!}

Combination (order doesn't matter):

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

10.3 Basic Probability

Probability:

P(A) = \frac{|A|}{|\Omega|}

(for equally likely outcomes) Conditional probability:

P(A|B) = \frac{P(A \cap B)}{P(B)}

Bayes' theorem:

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

✅ Practice Questions

Q1: How many ways to choose a committee of 3 from 10 people?

Solution

$\binom{10}{3} = \frac{10!}{3!7!} = 120$ ways. Q2: What's the probability of getting exactly 2 heads in 3 coin flips? Solution

Total outcomes: $2^3 = 8$ . Favorable: choose 2 of 3 flips to be heads = $\binom{3}{2} = 3$ . Probability = $3/8$ . Join Discord PreviousUncountable SetsNextGraph Theory Introduction