The language of probability

A trial is a procedure that can be repeated and gives one result. Examples include rolling a die once, selecting one card, or asking one student whether they travelled by bus. An outcome is one possible result of that trial. A sample space is the set of every possible outcome. In this syllabus, it is written as $U$, where $U$ is the sample space being discussed and has no unit.

An event is a subset of the sample space containing the outcomes of interest. If $A$ is an event, $n(A)$ gives the number of outcomes in event $A$. It is a count, so it has no unit. In the same way, $n(U)$ gives the number of outcomes in the sample space $U$ and is also a count with no unit.

When outcomes are equally likely, they all have the same chance of occurring. In this common special case,

The words equally likely matter. Counting sectors won’t give the probability if a spinner has unequal sectors. For a fair die, however, counting faces works. Probability lies between $0$ and $1$: $0$ represents an impossible event, $1$ represents a certain event, and values between them measure risk or likelihood.

Representing sample spaces

You can show a sample space as a list, a set, a table, or a grid. A list is enough for one die: $U=\{1,2,3,4,5,6\}$. With two trials, a table often helps prevent missed cases by making you account for every pair of outcomes.

Two fair dice sample space with at least one 6 highlighted.

A relative frequency is the proportion observed after repeated trials. If an event occurs $h$ times in $N$ trials, its relative frequency is

.

This is experimental probability because it comes from data. Theoretical probability comes from a model, such as the assumption that a coin is fair. After many repetitions, the relative frequency may settle near the theoretical probability, but it remains an approximation rather than a guarantee.

That makes simulations useful. A calculator or spreadsheet can simulate a random process thousands of times very quickly, so the connection between theory and observation becomes visible. Simulations also show that probability isn’t certainty: a $0.1$ chance does not mean exactly one occurrence in every ten trials.

Complementary events

The complement of an event contains every outcome in the sample space that is not in the original event. The complement of $A$ is written $A'$, where $A'$ is the event that $A$ does not occur.

Either $A$ occurs or $A'$ occurs, but they cannot both occur. Therefore,

This is often the quickest method for phrases such as "at least one". Rather than listing every possible way to get at least one success, calculate $1-$ the probability of getting none.

Expected number of occurrences

The expected number of occurrences is the long-run average count that a probability model predicts over a fixed number of trials. Suppose an event has probability $p$, where $p$ is the probability of the event occurring in one trial and is dimensionless. If there are $N$ trials, where $N$ is the number of trials and is a count with no unit, then

For example, suppose a school expects a $0.08$ probability that a student will be late on a wet morning. Among $250$ students, the expected number who are late is $250\times 0.08=20$. This does not require exactly $20$ students to be late; an expected value is an average produced by the model. That distinction matters when probability is used in insurance, planning, genetics, physics, or risk decisions, where people may perceive danger differently from how the numbers describe it.

Choosing the right representation

Probability questions are easier once you can see their structure. A Venn diagram is a set diagram showing events as regions within a sample space. Use one for overlaps such as $A$ and $B$, or for wording such as "or", "and", and "not".

A tree diagram is a branching diagram used to show a sequence of trials. It is particularly useful when probabilities change after the first outcome, especially in selection without replacement. A sample space diagram is a grid or table listing paired outcomes. It works well for two dice or two spinners, and for two independent selections where each pair can be counted.

When to use each probability diagram or table

Combined events and the meaning of "or"

Let $B$ be a second event in the same sample space. It is a subset of $U$ and has no unit. The event $A\cup B$ is the union of events $A$ and $B$, meaning $A$ or $B$ or both. In everyday English, "or" is often understood as exclusive. In probability, it is normally inclusive unless the question states otherwise.

The general addition rule is

Here, $P(B)$ is the probability that event $B$ occurs. The event $A\cap B$ means that both $A$ and $B$ occur, while $P(A\cap B)$ is the probability of that intersection. All these probabilities are dimensionless.

The overlap is already counted once in $P(A)$ and again in $P(B)$, so $P(A\cap B)$ must be subtracted. This is the safe formula to use unless the question gives a reason to simplify it.

Mutually exclusive events

Two events are mutually exclusive events when they cannot occur in the same trial. In symbols,

For mutually exclusive events, the general addition rule becomes

Don’t confuse mutually exclusive events with independent ones. If two non-impossible events are mutually exclusive, the occurrence of one tells you that the other did not occur. They are therefore not independent.

Conditional probability

A conditional probability is a probability calculated after some information is known. The notation $P(A\mid B)$ means the probability that $A$ occurs given that $B$ has occurred, and $P(A\mid B)$ is dimensionless. Its formal definition is

provided $P(B)\ne 0$. Rearranging gives

The word "given" is a useful warning light. Once you know that $B$ has happened, the sample space is no longer all of $U$; it is now the part inside $B$. On a Venn diagram, compare the overlap $A\cap B$ with the whole of $B$, not with the entire rectangle.

With and without replacement

In selection problems, with replacement means that an item is returned before the next selection, restoring the composition of the collection. Without replacement means that the item is not returned, so later probabilities usually change.

A tree diagram shows this difference clearly. Multiply branch probabilities along a path. When different paths satisfy the event, add their probabilities. If the tree includes every possible outcome, its terminal probabilities should add to $1$. This provides a useful built-in check.

Independent events

Two events are independent events when knowing that one has occurred does not change the probability of the other. For independent events,

Don’t assume this formula applies simply because two events appear unrelated. Independence is a condition to check, or information to use when the question states that the events are independent. The distinction matters in gambling contexts: probability can calculate risk clearly, but using it to exploit players raises an ethical question rather than a purely mathematical one.

The formal definition revisited

The formal definition of conditional probability isn’t new, but this syllabus point requires you to use it deliberately:

with $P(B)\ne 0$.

For tree diagrams, the rearranged form is often more useful:

A complete path probability comes from multiplying the first-branch probability by the conditional probability on the second branch.

If the given event is $B'$, use the companion version:

Independence using conditional probability

There are two equivalent ways to test for independence. The multiplication test is

The conditional-probability test is

If $A$ is independent of whether $B$ occurs, its probability stays the same inside $B$, outside $B$, and across the whole sample space. In applications such as medical studies or risk-factor analysis, this tests mathematically whether one classification appears to affect another.

Testing for independence in practice

When a question asks whether two events are independent, give a numerical comparison rather than just a sentence. Calculate both sides of one independence condition, then compare them.

Using the multiplication test, for example, calculate $P(A\cap B)$ and $P(A)P(B)$. Equal values show that the events are independent; unequal values show that they are dependent. Alternatively, calculate $P(A\mid B)$ and compare it with $P(A)$.

Make sure the conclusion matches the calculation. One common trap: mutually exclusive events aren’t automatically independent. If $P(A)>0$ and $P(B)>0$, mutually exclusive events have $P(A\cap B)=0$ but $P(A)P(B)>0$, so they fail the independence test.

Two-way table for independent events $A$ and $B$

Probability appears across many disciplines, but its interpretation depends on the context. A calculation may show dependence between two medical or social categories, yet it cannot explain the cause by itself. Here, the distinction between mathematical evidence and subject-specific reasoning is useful rather than artificial.

Reversing a conditional probability

Bayes' theorem combines prior and conditional probabilities to reverse the direction of conditioning. Put simply, it answers questions such as: if the test is positive, how likely is each possible cause?

Begin with two mutually exhaustive alternatives, $B$ and $B'$. Mutually exhaustive events have a union that covers the whole sample space, so one of them must occur. For two alternatives, Bayes' theorem is

The denominator gives the total probability of $A$. It splits this probability into the routes through $B$ and $B'$. Students often miss this step: Bayes is simply conditional probability combined with a complete list of routes to the evidence.

Bayes' theorem for three events

When there are a maximum of three possible underlying events, label them $B_1$, $B_2$, and $B_3$. Here, $B_i$ represents the $i$th possible underlying event for $i=1,2,3$. Together, the events should partition the sample space, meaning exactly one occurs.

Bayes' theorem becomes

A tree diagram usually provides the clearest working method. Place the three possible sources, groups, machines, diagnoses, or categories on the first set of branches, then put the observed event $A$ and its complement on the second set. Multiply along each path. Finally, divide the path you want by the total of all paths ending in $A$.

Link with independence

Bayes' theorem is much less interesting when $A$ is independent of the underlying event. If $A$ is independent of $B_i$, then $P(A\mid B_i)=P(A)$ for each relevant $i$. Observing $A$ therefore doesn’t update the probabilities of the $B_i$ events. This is the SL independence idea in a new setting.

Base rates matter when Bayes' theorem is used in medical screening, economics, quality control, and other real systems. If false positives are common, a rare condition may still be unlikely after a positive test. The mathematics is applicable, but its value depends on whether the model is valid and the probabilities are reliable.