Home / IB DP Maths / Application and Interpretation HL / IB Mathematics AI SL Concepts of trial, outcome, equally likely outcomes MAI Study Notes

IB Mathematics AI SL Concepts of trial, outcome, equally likely outcomes MAI Study Notes - New Syllabus

IB Mathematics AI SL Concepts of trial, outcome, equally likely outcomes MAI Study Notes

LEARNING OBJECTIVE

  • Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event

Key Concepts: 

  • Probability & Types of Events

MAI HL and SL Notes – All topics

Trial, Outcome, and Equally Likely Outcomes

In probability, we often deal with experiments or activities that produce random results. These are called trials. The result of a trial is called an outcome.

If all the outcomes of a trial have the same chance of occurring, they are said to be equally likely.

Consider the experiment of tossing a fair six-sided die.

The trial is rolling the die once.
The possible outcomes are: 1, 2, 3, 4, 5, and 6.
Since the die is fair, each number has an equal probability of $\frac{1}{6}$.

Example

A spinner is divided into 4 equal sectors labelled A, B, C, and D.

What is the probability of the pointer landing on:

a) Sector B?
b) A sector other than A?

▶️Answer/Explanation

Solution:

Total outcomes = 4 (A, B, C, D)
All outcomes are equally likely.

a)
Favorable outcome = B

$
P(\text{B}) = \frac{1}{4}
$

b)
Favorable outcomes = B, C, D

$
P(\text{not A}) = \frac{3}{4}
$

Relative Frequency

Relative frequency is used when we perform an experiment multiple times. It tells us how often a particular outcome occurs relative to the total number of trials.

It is calculated using the formula:

$\text{Relative Frequency} = \frac{\text{Number of times the event occurs}}{\text{Total number of trials}}$

Example

Using the concept of Relative frequency,

Suppose a coin is tossed 100 times and it lands on heads 57 times.

The relative frequency of getting heads is:

▶️Answer/Explanation

Solution:

$\frac{57}{100} = 0.57$

This means that based on the experiment, the probability of getting heads is estimated as 0.57.

Example

A die is rolled 200 times and the number 3 appears 42 times.

Estimate the probability of getting a 3 using relative frequency.

▶️Answer/Explanation

Solution:

$
\text{Relative Frequency} = \frac{\text{Number of times 3 appears}}{\text{Total trials}} = \frac{42}{200} = 0.21
$

So, estimated probability of getting a $3 = 0.21$

Sample Space and Event

The sample space, usually represented by the symbol $U$, is the set of all possible outcomes of an experiment.

An event is any collection of outcomes from the sample space. It can include one or more outcomes.

Let’s say two coins are tossed.

The sample space is:
$U = \{HH, HT, TH, TT\}$

Let A be the event “exactly one head appears”.

Then $A = \{HT, TH\}$

Example

Two coins are tossed together.

a) List the complete sample space.
b) Find the probability of getting exactly one head.
c) Find the probability of getting at least one tail.

▶️Answer/Explanation

Solution:

a)
Sample space $U = \{HH, HT, TH, TT\}$

b)
Exactly one head = HT or TH → 2 outcomes

$
P(\text{1 head}) = \frac{2}{4} = \frac{1}{2}
$

c)
At least one tail = HT, TH, TT → 3 outcomes

$
P(\text{≥1 tail}) = \frac{3}{4}
$

Probability of an Event

The probability of an event $A$ happening is the ratio of the number of favorable outcomes to the total number of possible outcomes in the sample space.

The formula is:

$
P(A) = \frac{n(A)}{n(U)}
$

where:

$n(A)$ is the number of favorable outcomes
$n(U)$ is the total number of outcomes

Example

What is the probability of drawing a heart from a standard deck of 52 playing cards?

$P(\text{heart})$

Hint: There are 13 hearts in a deck.

▶️Answer/Explanation

Solution:

There are 13 hearts in a deck.

So, $P(\text{heart}) = \frac{13}{52} = \frac{1}{4}$

Example

A bag contains 5 red balls, 3 green balls, and 2 blue balls.
One ball is picked at random.

What is the probability that the ball is:

a) Red?
b) Not green?

▶️Answer/Explanation

Solution:

Total balls = 5 + 3 + 2 = 10

a)
Red balls = 5

$
P(\text{Red}) = \frac{5}{10} = \frac{1}{2}
$

b)
Not green = red or blue = 5 + 2 = 7

$
P(\text{Not green}) = \frac{7}{10}
$

Complementary Events

Two events are complementary if one event happening means the other cannot happen, and vice versa. The complement of event A is written as $A’$, which means “not A”.

The total probability of an event and its complement is always 1:

$
P(A) + P(A’) = 1 \quad \Rightarrow \quad P(A’) = 1 – P(A)
$

Example

Let event A be “rolling a 6” when rolling a fair die.

What is probability of not getting a 6

$P(A) + P(A’) = 1$

▶️Answer/Explanation

Solution:

$P(A) = \frac{1}{6}$
So, $P(A’) = 1 – \frac{1}{6} = \frac{5}{6}$
This means the probability of not getting a 6 is $\frac{5}{6}$.

Example

The probability of it raining tomorrow is 0.35.

What is the probability that it will not rain tomorrow?

▶️Answer/Explanation

Solution:

$
P(\text{Not raining}) = 1 – P(\text{Raining}) = 1 – 0.35 = 0.65
$

Expected Number of Occurrences

The expected number of occurrences tells us how many times an event is likely to happen when a random experiment is repeated multiple times.

It is not always a whole number, and it does not guarantee the actual number of occurrences—it is a prediction based on probability.

$\text{Expected Number of Occurrences} = \text{Probability of the event} \times \text{Number of trials}$

Example

A die is rolled 60 times.

What is the expected number of times a 4 will appear?

Hint: $\text{Expected Number of Occurrences} = \text{Probability of the event} \times \text{Number of trials}$

▶️Answer/Explanation

Solution:

Probability of getting a 4 on a fair die $=\frac{1}{6}$
Number of trials $=60$

$\text { Expected occurrences of } 4=\frac{1}{6} \times 60=10$

 So, we expect the number 4 to appear 10 times out of 60 rolls.

Experimental vs Theoretical Probability

Experimental Probability

Experimental probability is determined by conducting an actual experiment and recording the outcomes.

Example

A coin is tossed 10 times:

Heads = 7 times
Tails = 3 times

Then:

$P(\text{head})$

▶️Answer/Explanation

Solution:

$P(\text{head}) = \frac{7}{10}, \quad P(\text{tail}) = \frac{3}{10}$

This type of probability is based on real-life data or observation.

Theoretical Probability

Theoretical probability is based on mathematical reasoning and assumes all outcomes are equally likely.

 

Example

A fair coin is tossed once

$P(\text{head})$

Mention all the sample space.

▶️Answer/Explanation

Solution:

Favorable outcomes for head = 1
Total possible outcomes = 2 (Head, Tail)

Then:

$P(\text{head}) = \frac{1}{2}, \quad P(\text{tail}) = \frac{1}{2}$

This type of probability is used when no actual experiment is conducted.

Scroll to Top