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Counting Methods

Under the classical view, a probability is a ratio of counts:

Probability by counting
P(A)=∣A∣∣Ω∣P(A) = \frac{|A|}{|\Omega|}

When outcomes are equally likely a probability is two counts and a division, and the difficulty is entirely in the counting.

So most probability questions are really counting questions, and counting has one decision at its centre: does order matter? Answer that correctly and the formula follows. Answer it wrongly and every subsequent step is wasted.

Permutations: order matters

Arranging nn distinct items in a sequence:

n!=n×(n−1)×⋯×2×1n! = n \times (n-1) \times \cdots \times 2 \times 1

Choosing rr of nn and arranging them:

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

Think of it as filling rr slots in order: nn choices for the first, n−1n-1 for the second, and so on. Podium finishes, orderings of trades, and passwords are permutations, because gold-silver-bronze is a different outcome from bronze-silver-gold.

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