Use sample spaces, events, complements, and expected frequencies

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Statistics and Probability

IB Mathematics: Analysis and Approaches SL — Study topic SL 4.5

A random experiment has an outcome that cannot be known with certainty in advance, even when the possible outcomes are known. Examples include rolling a die or selecting a coloured counter from a bag. To reason clearly about such situations, list the possible outcomes, describe the event of interest, and connect its probability to how often it might occur in repeated trials. This lesson uses only these ideas. The notation and counting methods rely on basic arithmetic and fractions.

What you will learn

Describe a random experiment using a sample space and identify events within it.
Calculate probabilities by counting outcomes when they are equally likely.
Use the complement of an event to find its probability.
Find and interpret the expected frequency of an event over repeated trials.

1. From outcomes to a sample space

A sample space is the set of all possible outcomes of a random experiment. It is usually written as

S

. Each individual outcome must be described precisely enough that the outcomes do not overlap and none are missing. For example, when a fair six-sided die is rolled once, the sample space is

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

.

An event is a collection of outcomes from the sample space. Events are often named with capital letters. If

A

is the event that the die shows an even number, then

A=\{2,4,6\}

. An outcome belongs to the event if it satisfies the stated condition.

When all outcomes are equally likely, the probability of an event can be found by counting. The number of outcomes in a set

A

is written as

n(A)

. The probability is the number of favourable outcomes divided by the total number of possible outcomes. This counting rule depends on equal likelihood; it cannot be used just because the outcomes have been listed.

P(A)=\frac{n(A)}{n(S)}

A sample space lists every possible outcome of the experiment.
An event is a subset of the sample space.
Use a favourable-outcomes fraction only when the outcomes are equally likely.

2. Complements and representations

The complement of event

A

, written A' in this lesson, means that

A

does not occur. Its outcomes are all the outcomes in the sample space that are not in

A

. An event and its complement cover the whole sample space without overlapping, so their probabilities add to

1

.

This gives a useful way to find the probability of an event that is awkward to count directly: count or calculate the outcomes in which it does not happen, then subtract that probability from

1

. For example, “at least one” often has the complementary description “none”.

The same situation can be represented in different ways. A list or table makes outcomes visible; a fraction shows the probability numerically; a set description identifies the event symbolically; and a context explains what the event means. A probability lies between

0

and

1

:

0

means impossible and

1

means certain. A probability is not a guarantee about one trial.

P(A)+P(A')=1

The complement includes every outcome in $S$ that is not in the event.
Use $P(A')=1-P(A)$ .
A probability describes the chance of an event, not a promise about the next outcome.

3. Repeated trials and expected frequency

Suppose an event has probability

p

on each trial and the experiment is carried out

N

times under the same conditions. The expected frequency is the number of times the event would occur on average over many sets of

N

trials. Calculate it by multiplying the number of trials by the probability.

Expected frequency is an estimate of a long-run average, not a prediction that the observed count must equal the answer. For example, an expected frequency of

18

means that counts around

18

are reasonable over repeated groups of trials; one particular group may have fewer or more. The expected frequency can be a decimal even though an actual count must be a whole number.

A graphing calculator or spreadsheet can simulate repeated trials by generating random outcomes and counting how often the event occurs. Compare the simulated proportion with the calculated probability, or the simulated count with the expected frequency. A simulation is a numerical check, not a replacement for identifying the sample space and calculating the probability. Different runs can produce different counts.

E=Np

Expected frequency is probability multiplied by the number of trials.
The trials are understood to use the same probability for the event.
An expected count is not a guaranteed observed count.

4. A reliable problem-solving routine

First state what one trial consists of and identify its possible outcomes. Next, define the event in words and as a set or condition. Check whether the outcomes are equally likely before using a counting fraction. If counting the event directly is difficult, consider its complement. Finally, when repeated trials are involved, multiply the probability by the number of trials and interpret the result as an average.

For a technology check, use a random-number function that matches the sample space. For a fair six-sided die, for example, generate integers from

1

to

6

, repeat the experiment, and count outcomes in the event. Do not use a function that makes some outcomes more likely unless the real experiment does so. Explain the theoretical probability before comparing it with the simulation.

In an exam-style response, make the assumption of equal likelihood clear when it matters, show the probability calculation, and include units or context for an expected frequency. If the question asks for a probability, give a value between

0

and

1

; if it asks how many times, report a frequency and state that it is expected rather than certain.

Define the experiment and event before calculating.
Check assumptions before counting outcomes.
Distinguish a theoretical answer from a variable simulation result.

Outcomes for a coin toss and a four-sided spinner

Coin result	Spinner result	Outcome pair
Heads	1	$(H,1)$
Heads	2	$(H,2)$
Heads	3	$(H,3)$
Heads	4	$(H,4)$
Tails	1	$(T,1)$
Tails	2	$(T,2)$
Tails	3	$(T,3)$
Tails	4	$(T,4)$

Worked example

1. A die event and its complement

A fair six-sided die is rolled once. Let

A

be the event that the result is greater than

4

. Find

P(A)

and the probability of the complement.

List the outcomes
The die can show any one of the six numbers, and fairness means these outcomes are equally likely.
$S=\{1,2,3,4,5,6\}$
Identify the event
The numbers greater than $4$ are $5$ and $6$ , so there are two favourable outcomes out of six.
$A=\{5,6\},\quad P(A)=\frac{2}{6}=\frac{1}{3}$
Use the complement
The complement consists of all results that are not greater than $4$ . Its probability is also found by subtracting the event probability from $1$ .
$P(A')=1-\frac{1}{3}=\frac{2}{3}$

Answer: The probability of rolling a number greater than

4

is

\frac{1}{3}

, and the probability of its complement is

\frac{2}{3}

.

Check: The probabilities add to

1

, as the event and its complement together include every possible result.

Worked example

2. A two-part outcome and an event

A fair coin is tossed and a fair four-sided spinner labelled 1,2,3,4 is spun once. Each coin outcome is equally likely, each spinner outcome is equally likely, and the two results do not affect one another. Find the probability of getting tails and an odd number.

Represent the sample space
Each outcome records both the coin result and the spinner result. There are two coin outcomes paired with four spinner outcomes, giving eight equally likely pairs.
$S=\{(H,1),(H,2),(H,3),(H,4),(T,1),(T,2),(T,3),(T,4)\}$
Select favourable outcomes
The event requires tails and an odd spinner number. The matching pairs are $(T,1)$ and $(T,3)$ , so two of the eight outcomes satisfy the condition.
$A=\{(T,1),(T,3)\},\quad P(A)=\frac{2}{8}=\frac{1}{4}$

Answer: The probability of getting tails and an odd number is

\frac{1}{4}

.

Check: The table of eight pairs shows that exactly two satisfy both conditions; the counting fraction is based on equally likely pairs.

Worked example

3. Expected frequency in repeated trials

A game has probability

\frac{3}{10}

of awarding a prize on each play. What is the expected number of prizes in

80

plays?

Identify the probability and trial count
The event is winning a prize. Its probability is $\frac{3}{10}$ on each play, and the total number of plays is $80$ .
$p=\frac{3}{10},\quad N=80$
Calculate the expected frequency
Multiply the probability of a prize by the number of plays. The result is a frequency, so it is interpreted as an average number of prizes.
$E=Np=80\left(\frac{3}{10}\right)=24$

Answer: The expected frequency is

24

prizes.

Check: This does not guarantee exactly

24

prizes in one set of

80

plays; it is the average indicated by the probability over repeated sets.

Common mistakes and how to avoid them

Using the favourable-outcomes fraction without checking whether the outcomes are equally likely.

Correction: Confirm equal likelihood first. If outcomes have different chances, simply counting them does not give the probability.

Leaving outcomes out of the sample space or listing the same outcome more than once.

Correction: Describe each outcome clearly and check that every possible result appears exactly once.

Treating the expected frequency as the count that must occur.

Correction: Interpret it as a long-run average; actual counts in a particular set of trials can differ.

Finding the complement by counting only outcomes that seem unlikely.

Correction: The complement is precisely all outcomes in the sample space that are not in the event.

Lesson summary

A sample space lists all possible outcomes; an event selects outcomes that meet a condition.
For equally likely outcomes, probability is the favourable count divided by the total count.
An event and its complement have probabilities that add to $1$ .
Expected frequency is the event probability multiplied by the number of trials, and is not a guaranteed count.

Check your understanding

Question 1

A fair die is rolled once. What is the probability that the result is not a

6

?

$\frac{1}{6}$
$\frac{5}{6}$
$\frac{1}{2}$
$1$

Show answer and explanation

\frac{5}{6}

Five of the six equally likely results are not

6

, so the probability is

\frac{5}{6}

. Equivalently, subtract

\frac{1}{6}

from

1

.

Question 2

An event has probability

0.4

on each trial. What is its expected frequency in

50

trials?

$0.8$
$20$
$40$
$50$

Show answer and explanation

20

Multiply the number of trials by the event probability:

50(0.4)=20

. This is an average, not a guaranteed count.

Key terms

Random experiment: A process with an outcome that is not known with certainty before it happens.
Sample space: The set of all possible outcomes of a random experiment.
Event: A specified outcome or collection of outcomes from a sample space.
Complement: The event consisting of all outcomes in the sample space that are not in a given event.
Expected frequency: The average number of times an event is expected to occur in a stated number of trials.

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Published by DoAssignment. This reviewed lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 4.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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