Probability from first principles

Sample space and events

The sample space is the set of all possible outcomes of a random process. An event is any collection of those outcomes. When every outcome is equally likely, the probability of an event is a ratio of counts.

Drawing one card from a well-shuffled deck has 52 equally likely outcomes. The event “a heart” contains 13 of them, so its probability is 13/52 = 1/4.

Rules you will use constantly

• Probabilities lie between 0 and 1, and the probabilities of all outcomes add up to 1.
• Complement: P(not A) = 1 − P(A).
• Addition for mutually exclusive events: P(A or B) = P(A) + P(B). In general P(A or B) = P(A) + P(B) − P(A and B).
• Multiplication for independent events: P(A and B) = P(A) × P(B). Without replacement, events are usually not independent.

The 'at least one' trick

Counting every way to get one or more of something is tedious. It is usually easier to count the opposite, getting none, and subtract from 1.

Probability of at least one ace in a five-card draw: the number of five-card sets with no ace is C(48, 5) = 1,712,304 out of 2,598,960, about 65.88%. So at least one ace has probability 1 − 0.6588 ≈ 34.12%.

Conditional thinking without replacement

The probability that the second card is a heart depends on what the first card was. If the first was a heart, 12 hearts remain among 51 cards (12/51 ≈ 23.5%); if not, 13 remain (13/51 ≈ 25.5%). Averaging these two cases, weighted by how likely each is, returns exactly 13/52. Symmetry: before you look, every position in a shuffled deck is equally likely to hold any card.

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