Basic Probability Rules
Apply the complement rule, addition rule for mutually exclusive events, and the general addition rule P(A∪B) = P(A) + P(B) − P(A∩B). Use Venn diagrams and two-way tables to organize probability calculations.
A lesson of AP Statistics, in Probability, about 10 minutes.
Probability Basics
By the end of this lesson, you will be able to define probability, use basic rules (complement, addition) to compute event probabilities, and construct sample spaces for simple experiments.
What Is Probability?
Probability is a number between 0 and 1 that describes how likely an event is to occur.
- P = 0: impossible
- P = 1: certain
- P = 0.5: equally likely as not
Interpretation of probability:
Frequentist: the long-run relative frequency of an event in an infinite sequence of identical, independent trials. Rolling a fair die, the probability of getting a 3 is 1/6 — meaning in the long run, roughly 1/6 of all rolls will produce a 3.
Subjective (Bayesian): a personal degree of belief about the likelihood of an event.
Law of Large Numbers: as the number of independent trials increases, the empirical proportion of times an event occurs approaches its true probability.
Sample Space, Events, and Outcomes
Sample space (S): the set of all possible outcomes of a random phenomenon.
Event: any collection of outcomes from the sample space. An event can consist of one or many outcomes.
Outcome: a single possible result.
Example: Flip a fair coin twice.
Sample space: S = {HH, HT, TH, TT}
Event A = "at least one head" = {HH, HT, TH}
P(A) = 3/4
Basic Probability Rules
Rule 1: All probabilities are between 0 and 1. $$0 \leq P(A) \leq 1$$
Rule 2: The sum of all outcome probabilities = 1. $$\sum P(\text{outcomes}) = 1$$
Rule 3: Complement Rule. $$P(A^c) = 1 - P(A)$$
where $A^c$ ("A complement") is the event that A does not occur.
Example: P(it rains tomorrow) = 0.3, so P(it does not rain) = 1 − 0.3 = 0.7.
Rule 4: Addition Rule for Mutually Exclusive Events.
Events A and B are mutually exclusive (disjoint) if they cannot both occur at the same time.
$$P(A \cup B) = P(A) + P(B) \quad \text{if A and B are mutually exclusive}$$
Rule 5: General Addition Rule (any two events).
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
where $P(A \cap B)$ is the probability that both A and B occur (their intersection). Subtract it to avoid double-counting.