What Is Probability and How Do You Calculate It?

Probability is a number between 0 and 1 that measures how likely an event is to occur. You calculate it by dividing the number of favorable outcomes by the total number of equally likely outcomes in the sample space. This works cleanly for experiments like coin flips, dice rolls, and card draws, where every outcome has the same chance. When outcomes aren't equally likely, or when events combine, you apply the addition and multiplication rules instead. The sections below cover the definitions, the core formulas, and worked examples so you can solve basic probability problems yourself.

The basic definition

Probability assigns a value to an event based on how often it would occur over many repetitions.

• A probability of 0 means the event cannot happen.
• A probability of 1 means the event is certain.
• Everything else falls in between.

For an experiment with equally likely outcomes:

P(Event) = (number of favorable outcomes) / (total number of outcomes)

The sample space is the set of all possible outcomes. An event is any subset of the sample space — one outcome or several grouped together.

Sample space and events

Before you can count favorable outcomes, you need to know the full sample space.

For "roll an even number," the favorable outcomes are {2, 4, 6} — three of them out of six total, so P = 3/6 = 0.5.

Worked example: a single die

Problem: What is the probability of rolling a number greater than 4?

1. Sample space: {1, 2, 3, 4, 5, 6} — 6 outcomes.
2. Favorable outcomes: {5, 6} — 2 outcomes.
3. P = 2/6 = 1/3 ≈ 0.333.

Problem: What is the probability of rolling a 7?

No outcome in the sample space equals 7, so P = 0. The event is impossible.

The addition rule

Use this when you want the probability that event A or event B occurs.

General form: P(A or B) = P(A) + P(B) − P(A and B)

Mutually exclusive events cannot happen at the same time, so P(A and B) = 0 and the rule simplifies to:

P(A or B) = P(A) + P(B)

Example: Drawing a king or a queen from a standard deck. These are mutually exclusive (a card can't be both).

• P(king) = 4/52
• P(queen) = 4/52
• P(king or queen) = 4/52 + 4/52 = 8/52 ≈ 0.154

If the events overlap, you must subtract the overlap or you'll double-count it.

The multiplication rule

Use this when you want the probability that event A and event B both occur.

General form: P(A and B) = P(A) × P(B | A)

Independent events don't affect each other's chances, so P(B | A) = P(B) and the rule simplifies to:

P(A and B) = P(A) × P(B)

Example: Flipping two heads in a row with a fair coin. The flips are independent.

• P(heads) = 1/2 on each flip
• P(heads and heads) = 1/2 × 1/2 = 1/4 = 0.25

Example with dependence: Drawing two aces from a deck without replacement.

• P(first ace) = 4/52
• P(second ace | first was an ace) = 3/51
• P(both aces) = 4/52 × 3/51 ≈ 0.0045

The second probability changes because the deck now has one fewer ace and one fewer card.

Choosing the right rule

The complement rule is often the fastest path. If a problem asks for "at least one," it's usually easier to compute the probability of "none" and subtract from 1.

Practice and tools

The concepts above cover the fundamentals: sample spaces, equally likely outcomes, and the addition, multiplication, and complement rules. To build fluency, work through solved problems and check your arithmetic with a calculator.

Statistics and Probability (stattrek.com) offers free tutorials on probability fundamentals with pages of solved problems, plus online tools including a Probability Calculator, a Bayes Rule Calculator, and a Combinations and Permutations calculator. These are useful for verifying answers and handling larger counts where manual enumeration gets tedious.