Probability

Probability Basics: How Likely Is an Event?

Probability is the language we use to talk about uncertainty. Before any forecast, A/B test or risk estimate, there are three simple building blocks: the random experiment, the outcome and the event.

The building blocks

Random experiment

A random experiment is any process whose result can't be predicted with certainty in advance. Tossing a coin or rolling a die are the classic examples: before you do it, you can't know whether you'll get heads or tails, a 3 or a 6. Every random experiment has three things in common:

More than one possible result. If there is only one, there is nothing to predict.
Results you can list in advance. You know the possibilities before you start.
Chance decides. Which of the possible results happens depends on luck.

Outcome

An outcome is one single result of the experiment. A die has six possible outcomes: 1, 2, 3, 4, 5 and 6.

Event

An event is a group of one or more outcomes we care about. "Rolling an even number" is an event made of the outcomes 2, 4 and 6. "Rolling an odd number" is another event: 1, 3 and 5. Probability always belongs to an event: it tells us how likely that group of outcomes is.

Experiment	Possible outcomes	Example event
Toss a coin	Heads, tails	Getting heads
Roll a die	1, 2, 3, 4, 5, 6	Rolling an even number (2, 4, 6)

The six outcomes of a die roll, with the even numbers 2, 4 and 6 highlighted as the event rolling an even number, which has probability 3 divided by 6 equals 0.5

Events can also be related to each other. Two types come up again and again:

Type 1

Mutually exclusive events

They cannot happen at the same time.

Example: rolling a 1 and rolling a 2 on a single die roll.

Used in the addition rule

Type 2

Independent events

One event does not affect the probability of the other.

Example: two separate coin tosses.

Used in the multiplication rule

The probability of an event

Probability is a number from 0 to 1, or equivalently a percentage from 0% to 100%.

0 (0%) means the event cannot happen.
1 (100%) means the event is certain.
Close to 0, say 0.05, means it is unlikely. Close to 1, say 0.95, means it is very likely.
0.5 (50%) means it is just as likely to happen as not to happen.

A scale from 0 to 1 showing impossible at 0, 10 percent chance of rain at 0.1, a fair coin at 0.5, 90 percent chance of rain at 0.9 and certain at 1 — Chart 1 The probability scale, with everyday examples.

Knowing a probability helps with decisions under uncertainty. With a 10% chance of rain tomorrow you can plan the picnic with confidence. With 90%, you'd better find another day.

How to calculate it

When every outcome is equally likely, the probability of an event is the number of outcomes that count as success divided by the total number of possible outcomes. This is called classical probability.

P(event) = number of desired outcomes ÷ total number of possible outcomes

Watch out

This formula only works when all outcomes are equally likely, as with a fair coin or a fair die. For a loaded die, it would give the wrong answer.

Example: coin toss

A fair coin has two equally likely outcomes, and one of them is heads:

P(heads) = 1 ÷ 2 = 0.5 = 50%

Now take a trick coin with heads on both sides. Every toss shows heads, so P(heads) = 100% and P(tails) = 0%.

Example: die roll

To roll a 3 with a fair die, there is one favourable outcome out of six:

P(3) = 1 ÷ 6 ≈ 0.167 ≈ 16.7%

For the event "rolling an even number" there are three favourable outcomes, so P = 3 ÷ 6 = 0.5.

Common confusion: what 50% does and doesn't mean

Saying that P(heads) = 50% is not a promise about any short series of tosses. Ten tosses of a fair coin can easily give 4 heads and 6 tails, or 7 and 3. The 50% describes the long run: the more you toss, the closer the share of heads tends to get to one half.

Chart 2 Simulated tosses of a fair coin. The share of heads wobbles early on and settles near 50% as the number of tosses grows.

Probability notation

In textbooks and technical documents, probability is written in a compact form. P stands for probability, and capital letters such as A and B name events:

P(A) is the probability of event A, and P(B) the probability of event B.
A′ (read "A prime" or "not A") is the complement of A: the event that A does not occur.
For any event A: 0 ≤ P(A) ≤ 1.
If P(A) > P(B), A is more likely than B. If P(A) = P(B), they are equally likely.

Basic probability rules

The complement rule

An event either occurs or it doesn't, so the two probabilities always add up to 1. That gives us the complement rule: the probability that an event does not occur is 1 minus the probability that it does.

P(A′) = 1 − P(A)

If the chance of rolling a 6 is 1/6, the chance of not rolling a 6 is 1 − 1/6 = 5/6, about 83.3%. And if there is a 10% chance of rain tomorrow, there is a 90% chance it stays dry.

A bar from 0 to 1 split into rolling a 6 with probability 1 over 6 and not rolling a 6 with probability 5 over 6 — Chart 3 The complement rule: an event and its complement together fill the whole bar.

The addition rule

The addition rule applies to mutually exclusive events. The probability that either A or B occurs is the sum of their individual probabilities:

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

Example: the probability of rolling a 1 or a 2 is 1/6 + 1/6 = 2/6 = 1/3, about 33.3%.

A row of six die faces with the 1 marked as event A and the 2 marked as event B; the events cannot both happen on one roll, so the probability of A or B is 1 over 6 plus 1 over 6 equals 1 over 3 — Chart 4 Mutually exclusive events: one roll can't be both a 1 and a 2, so their probabilities simply add up.

If the events can overlap

When two events can happen together, adding their probabilities counts the overlap twice, so you subtract it once: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events there is no overlap, which is why the simple version works.

Independent events: the multiplication rule

The multiplication rule applies to independent events. The probability that both A and B occur is the product of their probabilities:

P(A and B) = P(A) × P(B)

Typical examples are dice and coins, where one outcome has no influence on the next.

Example: the probability of getting heads on two tosses in a row is 0.5 × 0.5 = 0.25, or 25%. The same logic works across different experiments: heads on a coin and a 6 on a die has probability 1/2 × 1/6 = 1/12, about 8.3%.

A probability tree of two coin tosses: each toss has probability 0.5 for heads and tails, so each of the four paths, such as heads then heads, has probability 0.25 — Chart 5 Two coin tosses as a probability tree. Multiplying along the path gives the probability of each combination.

Summary of the rules

Rule	Formula	Use it when
Complement	P(A′) = 1 − P(A)	You want the probability that an event does not happen
Addition	P(A or B) = P(A) + P(B)	A and B are mutually exclusive
Multiplication	P(A and B) = P(A) × P(B)	A and B are independent

Together, these rules let us describe and calculate probabilities for different types of relationships between events, which is exactly what comes up again and again in data analysis.

Key takeaway

Probability puts a number between 0 and 1 on how likely an event is. When outcomes are equally likely, divide the favourable outcomes by all possible outcomes. Three rules go a long way: the complement rule for "not A", the addition rule for mutually exclusive events and the multiplication rule for independent events. Data professionals use them to help others make informed decisions about uncertain events.