Math101 · Independent Ontario learning library
Math101learn.math101.caProbability
Probability quantifies uncertainty from impossible to certain and connects sample spaces, long-run frequency, counting, and decision making.
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,
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.
Experimental probability
Experimental probability uses observed frequency:
As independent trials accumulate under stable conditions, relative frequency often settles near theoretical probability. Short runs can vary widely; ten coin flips need not contain exactly five heads.
Complements
The complement $A^c$ means that $A$ does not occur:
Addition rule
For any events $A$ and $B$,
The intersection is subtracted because it was counted twice. If events are mutually exclusive, they cannot occur together and $P(A\cap B)=0$.
Multiplication and independence
For independent events, one event does not change the probability of the other:
Two fair coin flips are independent, so $P(\text{two heads})=(1/2)(1/2)=1/4$. Drawing two cards without replacement is not independent because the first draw changes the deck.
Conditional probability
Conditional probability restricts attention to cases where $B$ occurred:
The denominator changes from the entire sample space to event $B$. Tables and tree diagrams help preserve that new reference group.
Counting systematically
Tree diagrams, organized lists, and the fundamental counting principle prevent missed or duplicated outcomes. If one choice has $3$ options and a following independent choice has $4$, there are $3\times4=12$ ordered combinations.
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 $1$?
Related topics
Explore the idea
Probability sampler
Change one quantity at a time and connect what moves to Probability.
success other outcome
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
A fair six-sided die is rolled. What is the probability of an even result?
- There are 3 even outcomes.
- There are 6 equally likely outcomes.
- P(even) = 3/6 = 1/2.
End of lesson
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