Counting & Probability

Where this is used

"In how many ways can five people pick a leader and a deputy?" That's a permutation — order matters. "How many two-person committees can those five form?" That's a combination — order doesn't. Loot drop rates, insurance premiums, cancer survival estimates, the settings an AI model is tuned with: counting and probability are how the modern world puts a number on things it can't predict.

Start gently

Counting problems are the craft of counting carefully. If the order matters it is a permutation; if it does not, a combination. Start with a tree so you can see exactly what you are counting.

Key points

Permutations

The number of ways to choose r things out of n when the order you choose them in matters: nPr = n!/(n−r)!.

Combinations

The number of ways to choose r things out of n when the order does not matter: nCr = n!/(r!(n−r)!).

What Probability Means

When every outcome is equally likely, P = (number of outcomes you want) ÷ (total number of outcomes). Every probability sits between 0 and 1.

The Addition Rule

If two events A and B can never happen together, P(A or B) = P(A) + P(B). If they can overlap, subtract the overlap once: P(A∪B) = P(A) + P(B) − P(A∩B).

Flip the Question Around

"At least one success" is usually painful to count directly. Work out the chance that everything fails and subtract it from 1 instead.

See it drawn

Picking 2 of 3 people and lining them up (permutations, 3P2 = 6)

Ignore the order and AB is the same as BA, which leaves 3C2 = 3 combinations.

Jobs that use this

Actuary$200k

Game Designer$80k

AI Researcher (Probabilistic Models)$200k