Basic Probability: Sets, Events, and the Three Rules That Always Work

Probability measures how likely an event is. Formally: P(A) = (number of outcomes in A) / (total number of equally likely outcomes). That fraction is all of basic probability. The hard part is correctly counting outcomes when the problem description makes that tricky.

Sample space and events

The sample space S is the set of all possible outcomes. An event A is any subset of S. For rolling a fair die, S = {1,2,3,4,5,6}. The event 'rolling an even number' is A = {2,4,6}. P(A) = 3/6 = 1/2.

The three core rules

• Complement rule: P(not A) = 1 − P(A). Often it is easier to calculate the probability that something does NOT happen.
• Addition rule: P(A or B) = P(A) + P(B) − P(A and B). The subtraction removes the double-count of outcomes in both A and B. If A and B cannot both happen (mutually exclusive), P(A and B) = 0.
• Multiplication rule: P(A and B) = P(A) · P(B|A). If A and B are independent, P(B|A) = P(B), simplifying to P(A) · P(B).

Combinations and permutations

When order matters, use permutations: P(n,k) = n! / (n−k)!. When order does not matter, use combinations: C(n,k) = n! / (k! · (n−k)!). Most probability problems about choosing a committee, hand of cards, or group use combinations because the people chosen are what matters, not the order they are listed.

Worked example: lottery

A lottery draws 6 numbers from 1 to 49. P(match all 6) = 1 / C(49,6) = 1 / 13,983,816 ≈ 0.0000000715. C(49,6) = 49! / (6! · 43!) = 13,983,816. The combination formula handles the counting; the probability is 1 over that count since only one combination wins.

Conditional probability and independence

P(B|A) is the probability of B given that A has already happened. P(B|A) = P(A and B) / P(A). Two events are independent if knowing one happened does not change the probability of the other: P(B|A) = P(B). Independent events are not the same as mutually exclusive events — mutually exclusive events are maximally dependent (if one happens, the other cannot).

Probability problems are often where students first encounter the combination formula feeling abstract. Solvequill can show a worked counting argument with diagrams and narration to make the C(n,k) formula feel visual rather than memorized.