Probability Fundamentals

Probability measures how likely an outcome is by comparing the number of favorable outcomes to the total number of equally likely outcomes. This lesson builds that ratio from an explicit, carefully listed sample space, which is the safest way to avoid missing or double-counting outcomes.

What you will learn

• Define probability as favorable over total outcomes
• Build an explicit sample space for a simple experiment
• Compute probabilities for equally likely outcomes

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Defining probability

The probability of an event is defined as the number of favorable outcomes divided by the total number of outcomes, provided every outcome is equally likely to occur. Written as a formula, $P(\text{event})=\dfrac{\text{favorable outcomes}}{\text{total outcomes}}$. A probability is always a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.

This formula only works when the phrase "equally likely" genuinely applies. A fair six-sided die has 6 equally likely faces, and a fair coin has 2 equally likely sides, but not every counting setup is automatically equally likely — later lessons return to this caution in more detail.

Building an explicit sample space

The sample space is the complete list of every possible outcome of an experiment. For a single die roll, the sample space is $\{1,2,3,4,5,6\}$. For two dice rolled together, each outcome is an ordered pair such as $(3,5)$, meaning the first die shows 3 and the second shows 5; since each die has 6 faces and the two rolls are independent, the sample space has $6\times6=36$ ordered pairs.

Writing out the sample space explicitly, rather than trying to reason about it in one's head, prevents two common errors: forgetting an outcome entirely, or counting the same outcome more than once. For dice sums, this matters especially, since $(2,5)$ and $(5,2)$ both give a sum of 7 but are two separate outcomes in the 36-outcome sample space.

Computing probabilities from the sample space

Once the sample space is written out, computing a probability is a matter of counting: count how many outcomes in the sample space satisfy the event being asked about, and divide by the total number of outcomes in the sample space. This favorable count should always be checked against the explicit list, not estimated.

The final fraction should always be reduced to lowest terms, matching the convention used throughout this course for exact answers. For example, if 4 outcomes out of a sample space of 8 are favorable, the probability is $\dfrac{4}{8}=\dfrac{1}{2}$, not left as $\dfrac{4}{8}$.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Forgetting to reduce the resulting fraction to lowest terms.
• Building an incomplete sample space, such as listing dice sums without accounting for every ordered pair that produces them.
• Assuming outcomes are equally likely without checking, which breaks the favorable-over-total formula.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• Probability is the ratio of favorable outcomes to total outcomes, valid when every outcome is equally likely.
• Building the sample space explicitly, one outcome at a time, prevents missing or double-counting outcomes.
• Always reduce the resulting probability fraction to lowest terms.