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D2.2 · Calculate and represent theoretical probability

Learn to calculate and represent theoretical probability through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Data Management

MBF3C study topic D2.2

Probability describes how likely an event is to happen. Theoretical probability uses a model of what could happen, rather than results from repeated trials. For example, before rolling a fair number cube, we can use its six possible results to find the chance of rolling a 2. In this lesson, you will list possible outcomes, identify which ones match an event, and compare the counts. You will also practise expressing the same probability in different forms.

What you will learn

Describe outcomes and events in a probability situation.
Calculate theoretical probability when outcomes are equally likely.
Represent a probability as a fraction, decimal, or percent.
Explain what a calculated probability means.

1. Review: outcomes, events, and fair chances

An outcome is one possible result of an action. When a number cube is rolled, getting a 1 is one outcome. The sample space is the set of all possible outcomes. For a standard number cube, the sample space is 1, 2, 3, 4, 5, and 6.

An event is the result, or group of results, we are interested in. For example, the event “roll an even number” includes 2, 4, and 6. Being clear about the event matters: “roll an even number” is different from “roll a number greater than 4.”

Theoretical probability works from a model of the situation. It assumes the outcomes being counted are equally likely, meaning each outcome has the same chance of occurring. A fair number cube is designed so each face has the same chance of landing face up. If outcomes are not equally likely, simply counting them does not give the probability.

Before calculating, check what the situation allows. A coin has two outcomes, heads and tails, in a simple model. In a bag with different numbers of coloured counters, count the individual counters as possible results. The colours are not equally likely when the bag contains different amounts of each.

An outcome is one possible result; a sample space lists all possible results.
An event is the result or group of results being considered.
Count outcomes only when the model treats them as equally likely.

2. Calculate a theoretical probability

To calculate theoretical probability, count the outcomes in the event and compare that count with the number of outcomes in the whole sample space. The outcomes that match the event are called favourable outcomes. “Favourable” means that the outcome meets the event we are asking about; it does not mean that the outcome is personally preferred.

For a fair number cube, there are six equally likely outcomes. Three are even, so the probability of an even result is three out of six. A fraction is useful because it shows the favourable count over the total count. Reduce the fraction when possible, but keep the original counts clear while setting it up.

The probability of an event is found by dividing the number of favourable outcomes by the total number of equally likely outcomes.

A probability is always from 0 to 1, inclusive. A probability of 0 means the event cannot occur in the model. A probability of 1 means it is certain in the model. A value between 0 and 1 describes a chance between impossible and certain. The calculated value is a model-based chance; it does not promise that an event will occur in a small number of tries.

Theoretical probability comes from the possible outcomes in a model. If you roll a number cube several times and record what actually happens, those results describe experimental probability. This lesson focuses on calculating from possible outcomes, not collecting trial results.

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

Count favourable outcomes and all equally likely outcomes.
A probability cannot be less than 0 or greater than 1.
Theoretical probability comes from the possible outcomes in a model.

3. Represent one probability in different forms

The same probability can be represented as a fraction, a decimal, or a percent. A fraction shows the counts directly. To write a fraction as a decimal, divide its numerator by its denominator. To write a decimal as a percent, multiply by 100 and attach the percent sign.

For example, one half can also be written as 0.5 or 50%. These forms mean the same amount. Choose a form that suits the question, and label it clearly so the reader knows whether a number is a fraction, decimal, or percent.

A percent can help make comparisons familiar, but it does not change the underlying chance. A probability of 25% means 25 out of every 100 in the model, or one quarter. It does not mean the event must happen exactly once in every four attempts.

When representing an answer, retain enough precision for the context. Some fractions convert to repeating decimals, so a decimal may need to be rounded if one is requested. Do not change a decimal to a percent by merely adding a percent sign; first convert it to a value on a scale out of 100.

\frac{1}{2}=0.5=50\%

Fractions, decimals, and percents can represent the same probability.
Convert fractions to decimals by dividing.
Convert decimals to percents by multiplying by 100.

4. Use an organized count and interpret the result

For a small sample space, you can list the outcomes. For a larger or less obvious situation, arrange them in an organized list or table so that none are missed or counted twice. The total count must describe the possible results in the model, not just the results that seem interesting.

Read the event carefully and mark every outcome that fits it. Then compare the number marked with the total number. Finally, check whether the answer is between 0 and 1 and whether its size makes sense. If nearly all outcomes fit the event, the probability should be close to 1; if very few fit, it should be close to 0.

A calculated probability describes the chance for one selection or trial under the stated model. In a bag situation, for instance, the model should make clear what is in the bag and whether a counter is returned before another selection. Do not assume extra conditions that the question has not given.

Organize outcomes when a list is hard to track.
Check that every possible result is counted once.
Interpret the answer in relation to the stated model.

Probability forms for the fair number cube example

Representation	Value	What it shows
Fraction	$\frac{1}{3}$	One favourable outcome for every three equal parts
Decimal	$0.333\ldots$	The fraction written in decimal form
Percent	About 33%	About 33 out of every 100 in the model

Worked example

A fair number cube

A fair six-sided number cube has faces numbered 1 to 6. Find the theoretical probability of rolling a number greater than 4. Represent the answer as a fraction and a percent.

Identify the sample space
The cube has six possible outcomes, and the fair model treats them as equally likely.
$\{1,2,3,4,5,6\}$
Select the favourable outcomes
The outcomes greater than 4 are 5 and 6, so there are two favourable outcomes.
$\{5,6\}$
Set up and reduce the probability
There are two favourable outcomes out of six equally likely outcomes. Divide the numerator and denominator by 2 to write an equivalent, simpler fraction.
$\frac{2}{6}=\frac{1}{3}$
Convert to a percent
Divide 1 by 3 to get a repeating decimal, then multiply by 100. Rounded to the nearest whole percent, the result is 33%.
$\frac{1}{3}=0.333\ldots\approx 33\%$

Answer: The probability is

\frac{1}{3}

, or about 33% to the nearest whole percent.

Check: Two of the six faces meet the event, so the probability is greater than zero and less than one half. The result is reasonable.

Worked example

Choosing a coloured counter

A bag contains 3 red counters, 2 blue counters, and 5 yellow counters. One counter is chosen at random. Find the theoretical probability of choosing a blue counter as a fraction, decimal, and percent.

Count all counters
Each individual counter is one possible selection. Add the three colour counts to find the total number of equally likely counter selections.
$3+2+5=10$
Count favourable counters
The event is choosing blue. There are two blue counters, so two selections out of ten match the event.
$\frac{2}{10}$
Simplify the fraction
Divide both the numerator and denominator by 2. This changes the form of the fraction without changing its value.
$\frac{2}{10}=\frac{1}{5}$
Write decimal and percent forms
Divide 1 by 5 to get 0.2. Multiplying 0.2 by 100 gives 20%.
$\frac{1}{5}=0.2=20\%$

Answer: The theoretical probability of choosing blue is

\frac{1}{5}

, 0.2, or 20%.

Check: Two of the ten counters are blue. Since there are fewer blue counters than half the total, a probability of 20% is sensible.

Common mistakes and how to avoid them

Using only favourable outcomes as the denominator.

Correction: Use the total number of equally likely outcomes as the denominator and the favourable count as the numerator.

Counting colours instead of counters in a bag.

Correction: For a random counter selection, count the individual counters. Each counter is a possible selection, and colours with more counters have more chances.

Thinking that a probability of 20% guarantees one success in every five tries.

Correction: The probability describes the chance in the model. A short series of trials can produce a different pattern.

Writing a decimal value with a percent sign without converting it.

Correction: Multiply a decimal by 100 before writing it as a percent. For example, 0.2 is 20%, not 0.2%.

Lesson summary

List or count the possible outcomes in the stated model.
Identify how many outcomes match the event.
Divide the favourable count by the total count when outcomes are equally likely.
Represent the probability as a fraction, decimal, or percent, and check that it is between 0 and 1.

Check your understanding

Question 1

A fair coin is flipped once. What is the theoretical probability of tails?

$\frac{1}{2}$
$\frac{1}{3}$
$1$
$0$

Show answer and explanation

\frac{1}{2}

There are two equally likely outcomes, heads and tails. One of them is tails, so the probability is

\frac{1}{2}

Question 2

A box contains 4 green and 6 orange counters. One is selected at random. What is the probability of selecting green?

40%
60%
$\frac{4}{6}$
4%

Show answer and explanation

40%

Four of the ten counters are green, so

\frac{4}{10}=\frac{2}{5}=0.4=40\%

Question 3

A fair number cube is rolled. What is the probability of rolling a number less than 3?

$\frac{1}{3}$
$\frac{1}{2}$
$\frac{2}{3}$
$\frac{1}{6}$

Show answer and explanation

\frac{1}{3}

The outcomes less than 3 are 1 and 2. There are two favourable outcomes out of six, so

\frac{2}{6}=\frac{1}{3}

Key terms

Outcome: One possible result of an action or selection.
Sample space: The set of all possible outcomes in a probability situation.
Event: A result, or group of results, being considered.
Equally likely: Outcomes that have the same chance of occurring in the model.
Favourable outcome: An outcome that matches the event being considered.
Theoretical probability: A probability calculated from the possible outcomes in a model.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D2.2. It is a study resource, not an official curriculum publication.

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