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Introductory Statistics Unit 3: Probability

Probability measures how likely an event is in the long run. This unit gives you the rules for combining events.

Covers: Probability rules, conditional probability, independence, two-way tables

Study guide

Probability measures how likely an outcome is in the long run, from 0 (impossible) to 1 (certain). When outcomes are equally likely, P(A) = (outcomes in A)/(total outcomes). Simulation estimates probabilities by repeating a random process many times.

Basic rules: the complement rule P(not A) = 1 − P(A) is the fastest way to handle "at least one" questions. The general addition rule is P(A or B) = P(A) + P(B) − P(A and B), subtracting the overlap so it isn't counted twice. Mutually exclusive events can't happen together, so their overlap is 0.

Conditional probability restricts attention to cases where B happened: P(A | B) = P(A and B)/P(B). Rearranging gives the general multiplication rule, P(A and B) = P(A)·P(B | A). Tree diagrams organize multi-stage problems: multiply along branches and add the paths that lead to the outcome you want. Reversing a conditional (finding P(B | A) from P(A | B)) uses the same tree, which is the idea behind Bayes' rule.

Events are independent if knowing one happened doesn't change the probability of the other: P(A | B) = P(A), equivalently P(A and B) = P(A)·P(B). Don't confuse independent with mutually exclusive: mutually exclusive events with positive probabilities are always dependent.

Two-way tables make these calculations concrete: joint probabilities come from cells divided by the grand total, and conditional probabilities come from cells divided by a row or column total.

Drawing with replacement keeps each draw's probabilities the same, so the draws are independent. Drawing without replacement changes the probabilities after each draw. For n independent trials, each with success probability p, P(at least one success) = 1 − (1 − p)ⁿ.

Key ideas

• Every probability is between 0 and 1, and all outcomes together add to 1.
• Complement rule: P(not A) = 1 − P(A).
• Mutually exclusive events can't happen together, so P(A and B) = 0.
• Conditional probability restricts the sample space to the cases where B happened.
• Independent events don't change each other's probability.

Formulas and rules

Common mistakes

• Treating mutually exclusive and independent as the same thing.
• Reading P(A | B) as P(B | A).

Words to know

General addition rule

P(A or B) = P(A) + P(B) − P(A and B)

Conditional probability formula

P(A | B) = P(A and B) / P(B)

Independent events

P(A | B) = P(A), or P(A and B) = P(A)·P(B).

Mutually exclusive events

Can't happen together; with positive probability, they're dependent.

Complement rule

P(not A) = 1 − P(A)

Probability

The long-run relative frequency of an outcome, from 0 to 1.

Sample space

The set of all possible outcomes.

Law of large numbers

With many trials, the observed proportion approaches the true probability.

Simulation

Imitating a random process to estimate probabilities.

General multiplication rule

P(A and B) = P(A)·P(B | A)

Tree diagram

Branches for each stage; multiply along branches.

Joint probability

P(A and B), from a cell ÷ grand total in a two-way table.

Sample practice questions

21 more questions like these, from level 1 to 5, are in the practice set.