Cheatsheet: Probability foundations

The one idea

A probability is a number from 0 to 1. Three rules (complement, addition, multiplication) combine probabilities, and the only fine print is independence: multiply only when events do not influence each other.

The basics

Probability scale:  0 = impossible ... 0.5 = even ... 1 = certain.   Always in [0,1].
Two readings: long-run frequency (repeat many times) OR degree of belief (calibrated confidence).
Equally likely outcomes:  P(event) = favorable outcomes / total outcomes.
  P(even on a die) = 3/6 = 1/2     P(more than 4) = {5,6} = 2/6 = 1/3

The three rules

Rule	Formula	When	Worked number
Complement	P(not A) = 1 - P(A)	Always; best for “at least one”	At least one head in 2 flips = 1 - 1/4 = 3/4
Addition (OR)	P(A or B) = P(A) + P(B) - P(A and B)	Always; subtract the overlap	King or heart = 4/52 + 13/52 - 1/52 = 4/13
Multiplication (AND)	P(A and B) = P(A) x P(B)	Only if INDEPENDENT	Two heads = 1/2 x 1/2 = 1/4

At least one: P(at least one) = 1 - P(none).
Chains: five independent 90% steps -> 0.9^5 = about 0.59 (a coin flip's reliability).

Independence

Independent = one event tells you nothing about the other (separate coin flips).
Dependent  = one outcome changes the other's odds (draw without replacement).
Multiply directly ONLY for independent events; dependent -> conditional probability (next lesson).
Independent events have no memory (the gambler's fallacy is forgetting this).

Pitfalls to dodge

• Forgetting to subtract the overlap in the OR rule (double-counts).
• Multiplying dependent events as if independent (need conditional probability).
• The gambler’s fallacy (a fair coin owes you nothing after a streak).
• Reporting a probability outside [0,1].
• Confusing AND (both, multiply) with OR (either, add minus overlap).

Words to use precisely

• Sample space: the set of all possible outcomes.
• Event: a subset of the sample space you care about.
• Complement: everything where the event does not happen; P(not A) = 1 - P(A).
• Independent events: one gives no information about the other; lets you multiply for an AND.
• Mutually exclusive: cannot both happen at once; the overlap term in the OR rule is zero.