P(event) = 0
The event cannot happen.
Free Probability Guide
Basic Probability
Learn how to identify a sample space, count favorable outcomes, write probability as a fraction, simplify when necessary, and interpret how likely an event is.
Probability measures how likely an event is to happen. In basic probability, the central idea is to compare the outcomes you want with all of the equally likely outcomes that could occur.
The pictures may change—a bag of marbles, a die, a spinner, coins, number tiles, or playing cards—but the reasoning stays the same.
What Is Probability?
Probability describes how likely an event is to happen. A probability can be written as a fraction, decimal, or percent.
Every probability is between 0 and 1. A probability of 0 means the event is impossible. A probability of 1 means the event is certain.
P(event) =
1
2
The event and its opposite have the same chance.
0 < P(event) < 1
The event can happen, but it is not guaranteed.
P(event) = 1
The event must happen.
Think of probability as a measure of chance.
The closer a probability is to 1, the more likely the event is. The closer it is to 0, the less likely the event is.
Sample Spaces, Outcomes, and Events
Before calculating a probability, identify the possible outcomes.
An outcome is one possible result. The sample space is the complete set of possible outcomes. An event is the outcome or group of outcomes you are interested in.
| Term | Meaning | Example |
|---|---|---|
| Outcome | One possible result | Rolling a 4 |
| Sample space | Every possible result | {1, 2, 3, 4, 5, 6} |
| Event | The result or group being studied | Rolling an even number |
| Favorable outcome | An outcome that makes the event happen | 2, 4, and 6 |
The denominator comes from the entire sample space.
If a fair six-sided die is rolled, there are always six total outcomes: 1, 2, 3, 4, 5, and 6.
The Basic Probability Formula
When every outcome is equally likely, probability is found by comparing the number of favorable outcomes with the total number of possible outcomes.
P(event) =
favorable outcomes
total possible outcomes
The numerator counts the outcomes that make the event happen. The denominator counts every possible outcome.
Favorable Outcomes
Count only the outcomes that satisfy the event.
Total Outcomes
Count every possible equally likely outcome.
For example, suppose we roll a fair six-sided die and want an even number.
Favorable outcomes:
2, 4, 6
Total outcomes: 1, 2, 3, 4, 5, 6
Total outcomes: 1, 2, 3, 4, 5, 6
P(even) =
3
6
=
1
2
Do not reverse the fraction.
Favorable outcomes belong on top. Total possible outcomes belong on the bottom.
Visual Basic Probability Examples
Probability problems can look very different, but the reasoning stays the same. Identify the event, count the favorable outcomes, and compare that count with the entire sample space.
These examples use the same visual style as the interactive Basic Probability tool.
Marbles
Find the probability of choosing a red marble.
3 red marbles out of 6 total:
P(red) = 3/6 = 1/2
Die
Find the probability of rolling an even number.
Even outcomes are 2, 4, and 6:
P(even) = 3/6 = 1/2
Coin
Find the probability of getting heads.
H
Heads
T
Tails
1 favorable outcome out of 2 total:
P(heads) = 1/2
Spinner
Find the probability of landing on purple.
2 purple sections out of 4 equal sections:
P(purple) = 2/4 = 1/2
Number Tiles
Find the probability of choosing an odd number.
1
2
3
4
5
6
7
8
9
10
Odd outcomes are 1, 3, 5, 7, and 9:
P(odd) = 5/10 = 1/2
Playing Cards
Find the probability of choosing a club.
4 clubs out of 10 cards:
P(club) = 4/10 = 2/5
The experiment changes, but the structure does not.
Whether you are looking at marbles, dice, coins, spinner sections, number tiles, or playing cards, the basic question is still: how many favorable outcomes are there out of how many total outcomes?
Simplifying Probability Fractions
After writing the probability fraction, check whether the numerator and denominator share a common factor greater than 1.
If they do, divide both numbers by the same factor.
4
10
=
2
5
In this example, both 4 and 10 are divisible by 2.
Original probability
4
10
Divide both by 2
4 ÷ 2 = 2
10 ÷ 2 = 5
10 ÷ 2 = 5
Simplified probability
2
5
If the fraction is already simplified, stop.
For example, 3 8 has no common factor greater than 1. There is no additional simplification step.
Simplifying does not change the probability.
The fractions 4/10 and 2/5 represent the same amount. Simplifying only gives the probability in a cleaner equivalent form.
Understanding the Probability Scale
Calculating a probability tells you more than just a fraction. The size of the probability tells you how likely the event is to happen.
Probabilities closer to 0 are less likely. Probabilities closer to 1 are more likely.
0
1/2
1
Impossible
Equally Likely
Certain
| Probability | Description | Meaning |
|---|---|---|
| 0 | Impossible | The event cannot happen. |
| Between 0 and 1/2 | Unlikely | The event is less likely to happen than not happen. |
| 1/2 | Equally Likely | The event and its opposite have the same chance. |
| Between 1/2 and 1 | Likely | The event is more likely to happen than not happen. |
| 1 | Certain | The event must happen. |
1
6
One favorable die face out of six is less than 1/2, so the event is unlikely.
5
6
Five favorable outcomes out of six is greater than 1/2, so the event is likely.
One-half is the key comparison point.
If a probability is below 1/2, the event is generally unlikely. If it is above 1/2, the event is generally likely.
Worked Basic Probability Examples
Each problem follows the same basic process: identify the event, count the favorable outcomes, count the total outcomes, write the probability, and simplify only when necessary.
4 blue marbles and 6 red marbles
What is the probability of choosing a blue marble?
- The event is choosing a blue marble.
- There are 4 favorable outcomes.
- There are 10 total marbles.
- Write the probability: 4 10
- Simplify: 4 10 = 2 5
Roll a number greater than 4
What is the probability of rolling a number greater than 4 on a fair six-sided die?
- The sample space is {1, 2, 3, 4, 5, 6}.
- The favorable outcomes are 5 and 6.
- There are 2 favorable outcomes and 6 total outcomes.
- Write the probability: 2 6
- Simplify: 2 6 = 1 3
Choose an even number from 1 through 8
Eight tiles are numbered 1 through 8. What is the probability of choosing an even number?
- The favorable outcomes are 2, 4, 6, and 8.
- There are 4 favorable outcomes.
- There are 8 total outcomes.
- Write the probability: 4 8
- Simplify: 4 8 = 1 2
3 clubs among 8 displayed cards
What is the probability of randomly choosing a club?
- The event is choosing a club.
- There are 3 favorable cards.
- There are 8 total cards.
- Write the probability: 3 8
- 3 and 8 have no common factor greater than 1, so 3/8 is already simplified.
Notice the repeated structure.
The objects change from problem to problem, but every example is built from the same comparison: favorable outcomes divided by total possible outcomes.
Common Basic Probability Mistakes
Most introductory probability errors come from counting the sample space incorrectly or building the probability fraction incorrectly.
1. Reversing the Probability Fraction
A student may accidentally put the total number of outcomes on top.
P(event) =
favorable
total
Favorable outcomes belong in the numerator. Total possible outcomes belong in the denominator.
2. Counting Only the Favorable Outcomes
If a bag contains 3 red marbles and 5 blue marbles, the probability of red is not 3/5. There are 8 total marbles, so the probability is 3/8.
3. Forgetting Repeated Outcomes
If a spinner contains four equal sections and two of them are purple, both purple sections count as separate favorable outcomes.
4. Simplifying When Nothing Can Be Simplified
A probability such as 3/8 is already in simplest form. Once the numerator and denominator have no common factor greater than 1, you are finished.
5. Forgetting to Simplify When Needed
Suppose 6 of 10 equally likely outcomes are favorable.
6
10
=
3
5
The original fraction is correct, but 3/5 is the simplified form.
6. Assuming Every Visual Outcome Is Equally Likely
Counting outcomes directly works when those outcomes are equally likely. For example, equal-sized spinner sections can be counted directly. If spinner sections have different sizes, their probabilities depend on the portion of the spinner they occupy.
A useful final check:
A probability can never be less than 0 or greater than 1. If your fraction is greater than 1, check whether you accidentally reversed the numerator and denominator or miscounted the sample space.
A Reliable Basic Probability Strategy
When you are not sure how to begin a basic probability problem, use the same sequence every time.
Step 1: Identify the Event
State exactly what outcome or group of outcomes
the problem is asking about.
Step 2: Identify the Sample Space
Determine every possible equally likely
outcome.
Step 3: Count Favorable Outcomes
Count the outcomes that make the event happen.
Step 4: Count Total Outcomes
Count all outcomes in the complete sample
space.
Step 5: Build the Probability
Write:
favorable
total
Step 6: Simplify If Needed
Reduce the fraction only if the numerator and
denominator have a common factor greater than
1.
Step 7: Interpret the Result
Decide whether the event is impossible,
unlikely, equally likely, likely, or certain.
A simple pattern to remember:
event → favorable → total → fraction →
simplify if needed → interpret
Practice What You Learned
Try the Basic Probability Interactive Tool
Practice probability with visual sample spaces using marbles, dice, coins, spinners, number tiles, and playing cards.
Work through favorable outcomes, total outcomes, probability fractions, simplification when needed, and probability interpretation step by step.
Start Basic Probability PracticeRelated Math Resources
Probability depends heavily on fractions, ratios, and careful counting. These resources can help build those supporting skills.
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