Sample Spaces & Basic Probability

Q: Can a probability be greater than 1?

No. A probability is always between $0$ and $1$, because the favorable outcomes are always part of the total outcomes. If your fraction comes out larger than $1$, the denominator is too small — check that it counts the entire sample space.

Q: Should I simplify my probability fraction?

Yes, reduce it: $\dfrac{2}{6}$ becomes $\dfrac{1}{3}$. A decimal or percent is also fine when the problem uses them, since $\dfrac{1}{2}$, $0.5$, and $50\%$ all say the same thing.

The sample space is the set of all possible outcomes of an experiment. Flip a coin and the sample space is $\{H, T\}$ . Roll a number cube and it's $\{1, 2, 3, 4, 5, 6\}$ . Every probability question starts here, because you can't count favorable outcomes until you know what all the outcomes are.

Once the sample space is written down, basic probability is one fraction: the number of outcomes you want, over the number of outcomes possible. Everything else in this unit — conditional probability, the addition rule, two-way tables — is built on that single fraction.

The probability formula

For equally likely outcomes, $P(\text{event}) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}$ . The denominator counts everything in the sample space; the numerator counts only the outcomes that match the event you care about.

A probability is always between $0$ and $1$ . A probability of $0$ means the event is impossible, and $1$ means it is certain. If you ever compute a probability bigger than $1$ , your denominator is wrong — usually because you divided by only part of the sample space.

Outcome vs. event

An outcome is one single result: rolling a $5$ . An event is any collection of outcomes: rolling an even number is the event $\{2, 4, 6\}$ , which contains three outcomes. When a question says "greater than" or "at least," list the outcomes in the event first — then count them.

Listing before counting sounds slow, but it's what prevents the classic miss. "Greater than $4$ " on a number cube is $\{5, 6\}$ — two outcomes, not three, because $4$ itself doesn't count.

The denominator is the whole space

The most common mistake in basic probability is dividing by the wrong total. If a bag has $3$ red marbles and $5$ white marbles, $P(\text{red})$ is $\dfrac{3}{8}$ , not $\dfrac{3}{5}$ . The denominator is everything in the bag — favorable outcomes plus unfavorable ones — never just the other group.

Worked examples

Example 1: an even number on a number cube

A standard six-sided number cube is rolled once. Find $P(\text{even number})$ .

Write the sample space

\{1, 2, 3, 4, 5, 6\}

List the even outcomes

\{2, 4, 6\}

Write favorable over total

P = \dfrac{3}{6}

Simplify

P = \dfrac{1}{2}

Answer: $P = \dfrac{1}{2}$

Example 2: marbles in a bag

A bag has $3$ red and $5$ blue marbles. One marble is drawn at random. Find $P(\text{red})$ .

Count the total marbles

3 + 5 = 8

Count the red marbles

3

Write favorable over total

P = \dfrac{3}{8}

Answer: $P = \dfrac{3}{8}$

Example 3: a spinner

A spinner has $8$ equal sections, and $2$ of them are green. Find $P(\text{green})$ .

Count all equal sections

8

Count the green sections

2

Write the probability and simplify

P = \dfrac{2}{8} = \dfrac{1}{4}

Answer: $P = \dfrac{1}{4}$

Try one yourself

Common questions

Can a probability be greater than 1?

No. A probability is always between $0$ and $1$ , because the favorable outcomes are always part of the total outcomes. If your fraction comes out larger than $1$ , the denominator is too small — check that it counts the entire sample space.

What's the difference between an outcome and an event?

An outcome is one single result of the experiment; an event is a set of outcomes. Rolling a $3$ is an outcome. Rolling an odd number is an event made of the three outcomes $\{1, 3, 5\}$ .

Do the sections or objects have to be equally likely?

Yes — the formula favorable over total only works when every outcome has the same chance. A spinner with unequal sections or a weighted cube needs the measures of the sections instead, which is what geometric probability handles.

Should I simplify my probability fraction?

Yes, reduce it: $\dfrac{2}{6}$ becomes $\dfrac{1}{3}$ . A decimal or percent is also fine when the problem uses them, since $\dfrac{1}{2}$ , $0.5$ , and $50\%$ all say the same thing.

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Sample Spaces & Basic Probability

The probability formula

Outcome vs. event

The denominator is the whole space

Worked examples

Example 1: an even number on a number cube

Example 2: marbles in a bag

Example 3: a spinner

Try one yourself

Common questions

Keep learning

Geometric Probability

Permutations & Combinations

The Addition Rule

Conditional Probability & Independence