Probability

Probability quantifies uncertainty from impossible to certain and connects sample spaces, long-run frequency, counting, and decision making.

Open the full lesson →

Events are subsets of a sample space, and conditional probability changes the reference set. A sample-space rectangle contains two overlapping event regions and a small two-stage probability tree. The overlap marks joint occurrence, while branch labels show how later probabilities can depend on an earlier outcome.

Probability assigns a number from $0$ to $1$ to an event: $0$ is impossible, $1$ is certain, and values between describe uncertainty.

Outcomes, events, and sample spaces

An outcome is one possible result. The sample space $S$ contains every possible outcome. An event $A$ is a collection of outcomes of interest.

For one fair six-sided die, $S=\{1,2,3,4,5,6\}$. The event “roll an even number” is $A=\{2,4,6\}$.

Equally likely outcomes

When all outcomes are equally likely,

P(A)=\frac{\text{favourable outcomes}}{\text{total outcomes}}.

For the even-number event, $P(A)=3/6=1/2$. This shortcut fails when outcomes are not equally likely, such as a biased spinner with unequal sectors.

Addition rule

For any events $A$ and $B$,

P(A\cup B)=P(A)+P(B)-P(A\cap B).

The intersection is subtracted because it was counted twice. If events are mutually exclusive, they cannot occur together and $P(A\cap B)=0$.

Interpreting probability

A probability of $0.8$ does not promise an event on the next trial. It describes a model or long-run tendency. Evaluate how the probability was estimated, whether assumptions are reasonable, and whether the source has uncertainty or bias.

Common mistakes

Assuming outcomes are equally likely without justification. Use geometry, mechanics, or data to support the model.

Adding overlapping events without subtracting the intersection. Watch for double counting.

Confusing mutually exclusive and independent. Nontrivial mutually exclusive events cannot be independent.

Believing short runs must match theory exactly. Random variation is expected.

Quick self-check

Have I defined the sample space and event?
Are outcomes equally likely?
Do events overlap, exclude one another, or act independently?
Would a complement, table, or tree make counting safer?
Does the final probability lie from $0$ to $1$?