Probability Basics

Sample space , event A ⊆ , measure P(A) ∈ [0, 1].

The idea

Probability theory is the mathematics of chance: it assigns each thing that might happen a number between $0$ and $1$ measuring how likely it is. For the finite experiments in this topic, making this precise takes three objects.

First, the sample space $\Omega$ is the set of all possible outcomes, exactly one of which will occur. For one roll of a die, $\Omega = \{1, 2, 3, 4, 5, 6\}$.

Second, an event is any subset $A \subseteq \Omega$. Ordinary statements about chance translate into set operations: "an even number" is the subset $\{2, 4, 6\}$, "not $A

Third, the probability measure $\mathbb{P}$ assigns each event a number in $[0, 1]$, subject to two rules: $\mathbb{P}(\Omega) = 1$, since some outcome must occur, and $\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B)$ whenever $A$ and $B$ are disjoint. These rules determine how probabilities combine in a finite sample space.

When $\Omega$ is finite and its outcomes are equally likely, the rules force $\mathbb{P}(A) = \frac{|A|}{|\Omega|},$ so a probability question becomes a counting question.

Ways to work on it

• Walkthrough. The basic language and rules of probability, taught with die rolls and coin flips.
• Practice. Decide which probability assignments are valid and combine probabilities of disjoint events.
• Hardest. Compute the probability of an event when two dice are rolled.

Not sure where to start? Take the ten-question placement test.