Theoretical Probability

Theoretical probability describes what we expect to happen based on mathematics and the possible outcomes of an event.

Unlike experimental probability, we do not need to perform an experiment first.

For example, consider a fair six-sided die.

The possible outcomes are:

1, 2, 3, 4, 5, 6

Because the die is fair, each outcome is equally likely.

There is one way to roll a 6 out of six possible outcomes.

Therefore:

P(6) = 1/6

This is a theoretical probability because it is calculated from what we know about the possible outcomes rather than from observed experimental results.

Theoretical probability allows us to answer questions such as:

• What is the probability of flipping heads?
• What is the probability of rolling an even number?
• What is the probability of drawing a heart from a deck of cards?
• Which outcome is more likely?
• How many times should we expect an event to occur?

What Is Theoretical Probability?

Theoretical probability is the probability of an event calculated using mathematical reasoning about all possible outcomes.

When all outcomes are equally likely:

P(event) = number of favourable outcomes / total number of possible outcomes

For example, a fair coin has two possible outcomes:

Heads, Tails

There is one favourable outcome for heads.

Therefore:

P(heads) = 1/2

= 0.5

= 50%

No coin needs to be flipped to calculate this probability.

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Possible Outcomes

An outcome is one possible result of a probability experiment.

For a coin:

Heads or Tails

For a standard six-sided die:

1, 2, 3, 4, 5, 6

For a spinner labelled A, B, C and D:

A, B, C, D

Before calculating a theoretical probability, we need to identify all possible outcomes.

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The Sample Space

The sample space is the complete set of possible outcomes.

It is often written inside braces.

For a coin:

S = {H, T}

For a six-sided die:

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

For a spinner with four numbered sections:

S = {1, 2, 3, 4}

Identifying the sample space helps ensure that we count every possible outcome.

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Favourable Outcomes

A favourable outcome is an outcome that satisfies the event we are investigating.

Suppose a die is rolled and we want:

P(rolling an even number)

Sample space:

{1, 2, 3, 4, 5, 6}

Favourable outcomes:

{2, 4, 6}

There are:

3 favourable outcomes

out of:

6 possible outcomes

Therefore:

P(even) = 3/6

= 1/2

= 0.5

= 50%

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Equally Likely Outcomes

The basic theoretical probability formula works directly when all individual outcomes are equally likely.

For a fair die:

P(1) = P(2) = P(3) = P(4) = P(5) = P(6)

Each probability is:

1/6

For a fair coin:

P(heads) = P(tails)

Each probability is:

1/2

The word fair is important because it tells us that the relevant outcomes are equally likely.

When Outcomes Are Not Equally Likely

We cannot always simply count outcomes.

Imagine a spinner where:

• half the spinner is red
• one quarter is blue
• one quarter is green

There are three colour outcomes:

red, blue, green

But this does not mean:

P(red) = 1/3

The sections are not equal.

Instead:

P(red) = 1/2

P(blue) = 1/4

P(green) = 1/4

The probability depends on the size of each section.

This is why checking the assumptions of a probability model is important.

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Theoretical Probability Formula

For equally likely outcomes:

P(event) = favourable outcomes / total possible outcomes

Suppose we roll a standard die.

What is the probability of rolling a number greater than 4?

Possible outcomes:

1, 2, 3, 4, 5, 6

Favourable outcomes:

5, 6

Therefore:

P(number > 4) = 2/6

= 1/3

≈ 0.333

≈ 33.3%

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Worked Example: Rolling a 3

A fair six-sided die is rolled.

Sample space:

{1, 2, 3, 4, 5, 6}

Favourable outcome:

{3}

Therefore:

P(3) = 1/6

≈ 0.167

≈ 16.7%

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Worked Example: Rolling an Odd Number

Odd numbers on a die are:

1, 3, 5

Therefore:

P(odd) = 3/6

= 1/2

= 50%

Rolling an odd number has an even chance.

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Worked Example: Rolling Less Than 5

Numbers less than 5 are:

1, 2, 3, 4

There are four favourable outcomes.

Therefore:

P(number < 5) = 4/6

= 2/3

≈ 66.7%

This event is more likely to happen than not happen.

Impossible Events

An impossible event has no favourable outcomes.

For example:

What is the probability of rolling an 8 on a standard six-sided die?

There are:

0 favourable outcomes

out of:

6 possible outcomes

Therefore:

P(8) = 0/6

= 0

An impossible event has probability:

0 = 0%

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Certain Events

A certain event includes every possible outcome.

For example:

What is the probability of rolling a number less than 7?

Possible favourable outcomes:

1, 2, 3, 4, 5, 6

Therefore:

P(number < 7) = 6/6

= 1

= 100%

A certain event has probability:

1

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Probability Can Be Written in Different Forms

Suppose:

P(event) = 3/4

As a decimal:

3 ÷ 4 = 0.75

As a percentage:

0.75 × 100 = 75%

Therefore:

3/4 = 0.75 = 75%

All three forms describe the same probability.

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Simplifying Probability Fractions

Probability fractions should usually be simplified.

Suppose:

P(even number on a die) = 3/6

Divide numerator and denominator by 3:

3/6 = 1/2

Both are mathematically correct, but:

1/2

is the simplest form.

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Probability with Coloured Counters

A bag contains:

5 red counters

3 blue counters

2 green counters

Total:

5 + 3 + 2 = 10 counters

Assume each counter has an equal chance of being selected.

Then:

P(red) = 5/10 = 1/2

P(blue) = 3/10

P(green) = 2/10 = 1/5

Therefore:

red is the most likely colour

and:

green is the least likely colour

Probability with Spinners

Suppose a spinner has eight equal sections.

They are:

3 red

2 blue

2 green

1 yellow

Because all sections are equal:

P(red) = 3/8

P(blue) = 2/8 = 1/4

P(green) = 2/8 = 1/4

P(yellow) = 1/8

Notice:

P(blue) = P(green)

Therefore, blue and green are equally likely.

Red is the most likely.

Yellow is the least likely.

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The Importance of Equal Sections

Suppose another spinner has four sections labelled:

A, B, C, D

It might be tempting to say:

P(A) = 1/4

But we cannot know this simply from the four labels.

If the sections have different sizes, their probabilities are different.

The assumption:

"Each outcome has probability 1/4"

is only justified if the sections are equal or we otherwise know that the outcomes are equally likely.

Probability with Playing Cards

A standard deck contains:

52 cards

There are four suits:

• hearts
• diamonds
• clubs
• spades

Each suit contains:

13 cards

Therefore:

P = 13/52

= 1/4

= 25%

There are:

26 red cards

Therefore:

P(red card) = 26/52

= 1/2

= 50%

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Worked Example: Drawing an Ace

A standard deck contains four aces.

Therefore:

P(ace) = 4/52

= 1/13

≈ 7.7%

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Worked Example: Drawing a Face Card

The face cards are:

Jack, Queen, King

There are three face cards in each of four suits.

Number of face cards:

3 × 4 = 12

Therefore:

P(face card) = 12/52

= 3/13

≈ 23.1%

Complementary Events

The complement of an event is the event that it does not happen.

For example:

Event: rolling a 6

Complement: not rolling a 6

Since:

P(event) + P(not event) = 1

we can calculate:

P(not 6) = 1 − 1/6

= 5/6

This is often easier than counting all the other outcomes separately.

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Worked Example: Complement

Suppose:

P(rain) = 0.35

Then:

P(no rain) = 1 − 0.35

= 0.65

= 65%

The probabilities of complementary events always total:

1

or:

100%

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Probability of "Not"

The complement rule is particularly useful for questions containing:

not

For example:

A bag contains:

6 red
3 blue
1 green

Total = 10

P(red) = 6/10

Therefore:

P(not red) = 1 − 6/10

= 4/10

= 2/5

We could also count:

3 blue + 1 green = 4 non-red counters

which gives the same answer.

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Comparing Theoretical Probabilities

Suppose:

P(A) = 1/5

P(B) = 3/5

Since:

3/5 > 1/5

Event B is more likely.

Probabilities allow us to compare events mathematically rather than relying only on words such as "likely" or "unlikely."

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Worked Example: Which Is More Likely?

A fair die is rolled.

Event A

Rolling a number greater than 4.

Favourable outcomes:

5, 6

P(A) = 2/6 = 1/3

Event B

Rolling an even number.

Favourable outcomes:

2, 4, 6

P(B) = 3/6 = 1/2

Since:

1/2 > 1/3

Event B is more likely than Event A.

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Expected Frequency

Theoretical probability can be used to predict approximately how often an event should occur.

Expected frequency = theoretical probability × number of trials

Suppose a fair coin is flipped:

200 times

P(heads) = 1/2

Expected heads:

200 × 1/2 = 100

We would expect approximately:

100 heads

This does not guarantee exactly 100 heads.

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Worked Example: Expected Sixes

A fair die is rolled:

300 times

P(6) = 1/6

Expected number of sixes:

300 × 1/6

= 50

We would expect approximately 50 sixes.

The actual experimental result might be:

47, 52, 55, or another value.

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Theoretical vs Experimental Probability

These two ideas are closely related but different.

Theoretical Probability

Based on:

mathematical reasoning

Example:

For a fair coin:

P(heads) = 0.5

Experimental Probability

Based on:

observed results

Suppose 100 flips produce 56 heads:

P(heads) = 56/100

= 0.56

The two probabilities are not exactly the same.

That is normal.

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Comparing Theoretical and Experimental Results

Suppose a fair die is rolled 120 times.

Theoretically:

P(6) = 1/6

≈ 0.167

Expected sixes:

120 × 1/6 = 20

Suppose the experiment produces:

23 sixes

Experimental probability:

23/120

≈ 0.192

Theoretical probability:

0.167

Experimental probability:

0.192

The experimental result is slightly higher.

This difference can occur because of random variation.

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More Trials Usually Improve the Comparison

Imagine repeatedly flipping a fair coin.

The theoretical probability remains:

0.5

As the number of trials increases, the experimental probability often becomes closer to the theoretical probability.

This is related to the law of large numbers.

Assumptions in Theoretical Probability

Theoretical probability depends on a mathematical model.

That model often requires assumptions.

For example, when we say:

P(6) = 1/6

we are assuming:

• the die has six possible faces
• the die is fair
• each face is equally likely
• the die is rolled normally
• no outside factor systematically favours one face

If these assumptions are incorrect, the theoretical model may not describe the real situation accurately.

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Assumption: A Fair Coin

When we write:

P(heads) = 1/2

we usually assume the coin is fair.

But imagine a specially manufactured coin that lands heads much more frequently.

The mathematical model:

P(heads) = 1/2

might no longer be appropriate.

This shows why theoretical probability depends on the assumptions behind the model.

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Assumption: A Fair Die

For a standard fair die:

P(each face) = 1/6

But if the die is weighted, damaged, or shaped unevenly, the outcomes may not be equally likely.

In that situation, experimental data might provide useful evidence about the actual probabilities.

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Assumption: Random Selection

Suppose a bag contains:

4 red counters
6 blue counters

We calculate:

P(red) = 4/10

= 0.4

This assumes that every counter has an equal chance of being selected.

If someone looks into the bag and deliberately chooses a colour, the selection is no longer random.

The theoretical model no longer describes the process properly.

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Assumption: Replacement

Suppose a bag contains:

5 red counters
5 blue counters

Initially:

P(red) = 5/10

= 1/2

Suppose a red counter is selected and not replaced.

The bag now contains:

4 red
5 blue

Total = 9

The probability of red on the next selection becomes:

P(red) = 4/9

The probabilities changed because the contents changed.

When objects are replaced after each selection, probabilities can remain constant from trial to trial.

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Models Are Simplifications

A probability model is a mathematical description of a random situation.

Models are useful because they allow us to:

• calculate probabilities
• compare outcomes
• make predictions
• estimate expected frequencies

But models depend on assumptions.

For example:

"A coin has a 50% chance of heads."

is a model based on the assumption of a fair coin.

Real objects and real situations may not perfectly match ideal mathematical models.

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When Experimental Results Disagree Strongly

Suppose a die is theoretically fair.

P(6) = 1/6

It is rolled:

6,000 times

Suppose 6 appears:

2,000 times

Experimental probability:

2000/6000

= 1/3

≈ 33.3%

This is very different from:

1/6 ≈ 16.7%

A large and persistent difference after many trials might suggest that we should investigate:

• whether the die is fair
• whether the experiment was biased
• whether the data were recorded correctly
• whether the theoretical assumptions are appropriate

Theoretical and experimental probability can therefore be used together to evaluate a probability model.

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Theoretical Probability in Real Life

Many real-world probability problems are more complicated than coins and dice.

Theoretical models are used in:

Genetics

Predicting possible inheritance patterns.

Engineering

Estimating the probability of system failures.

Insurance

Modeling risks.

Games

Calculating possible outcomes and fairness.

Computer Science

Designing randomized algorithms and simulations.

Physics

Modeling processes involving uncertainty.

Weather and Climate Science

Probability models contribute to predictions about possible future conditions.

In real-world applications, theoretical models may be combined with large amounts of observed data.

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Probability Trees

For situations involving several stages, a probability tree can help organize possible outcomes.

For example, flipping a coin twice produces:

First flip: H or T

Second flip: H or T

Possible combined outcomes:

HH

HT

TH

TT

There are four equally likely outcomes.

Therefore:

P(two heads) = 1/4

P(one head and one tail) = 2/4 = 1/2

P(two tails) = 1/4

Listing Outcomes Systematically

When a problem has several stages, listing outcomes systematically prevents us from missing possibilities.

Suppose a coin is flipped and a die is rolled.

Coin outcomes:

H, T

Die outcomes:

1, 2, 3, 4, 5, 6

Combined outcomes:

H1, H2, H3, H4, H5, H6
T1, T2, T3, T4, T5, T6

Total outcomes:

12

Suppose we want:

P(heads and an even number)

Favourable outcomes:

H2, H4, H6

Therefore:

P(heads and even) = 3/12

= 1/4

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Checking Your Answer

A theoretical probability should always satisfy:

0 ≤ P(event) ≤ 1

If you calculate:

P(event) = 1.4

something has gone wrong.

Similarly:

P(event) = −0.2

cannot be correct.

Probability can never be:

• negative
• greater than 1
• less than 0%
• greater than 100%

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A Strategy for Theoretical Probability Problems

When solving a probability problem:

Identify the experiment

What is happening?

Identify the sample space

What are all possible outcomes?

Check whether outcomes are equally likely

Do not assume this automatically.

Identify favourable outcomes

Which outcomes satisfy the event?

Calculate

For equally likely outcomes:

P(event) = favourable outcomes / total outcomes

Simplify

Simplify fractions where appropriate.

Convert if needed

Express the answer as a fraction, decimal, or percentage.

Check

Make sure:

0 ≤ P ≤ 1

Interpret

Explain what the probability means in context.

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Worked Example: Full Solution

A bag contains:

7 red counters

5 blue counters

3 green counters

One counter is selected randomly.

Total outcomes

7 + 5 + 3 = 15 counters

Probability of red

P(red) = 7/15

≈ 46.7%

Probability of blue

P(blue) = 5/15

= 1/3

≈ 33.3%

Probability of green

P(green) = 3/15

= 1/5

= 20%

Comparison

Red is the most likely colour.

Green is the least likely.

Probability of not green

P(not green) = 1 − 1/5

= 4/5

= 80%

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Worked Example: Comparing Theory and Experiment

A fair spinner has five equal sections:

A, B, C, D, E

Theoretical probability of A:

P(A) = 1/5

= 0.20

The spinner is spun 100 times.

A occurs 23 times.

Experimental probability:

P(A) = 23/100

= 0.23

Comparison:

The theoretical probability is 0.20.

The experimental probability is 0.23.

The experimental result is slightly higher, which can reasonably occur because of random variation.

More trials would generally provide a more stable comparison.

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Common Mistakes

Using the formula when outcomes are not equally likely

Counting outcomes works directly only when each individual outcome has equal probability.

Forgetting some possible outcomes

Write the sample space when necessary.

Counting favourable outcomes incorrectly

Check exactly what the question asks.

Assuming "fair" and "random" mean the same thing

Random means the result cannot be known with certainty beforehand.

Fair means equivalent outcomes have equal probabilities.

Confusing theoretical and experimental probability

Theoretical probability comes from a mathematical model.

Experimental probability comes from observations.

Expecting theory and experiment to match exactly

Random variation is normal.

Forgetting to simplify fractions

For example:

3/6 = 1/2

Giving a probability greater than 1

Probabilities must lie between 0 and 1.

Assuming a model without stating its assumptions

For example, P(6) = 1/6 assumes a fair six-sided die.

Treating expected frequency as a guarantee

Expected frequency describes what we anticipate, not exactly what must happen.

Ignoring whether objects are replaced

Without replacement, probabilities can change between selections.

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Key Terms

Theoretical probability — Probability calculated mathematically from possible outcomes and an assumed probability model.

Experimental probability — Probability calculated from observed results.

Probability model — A mathematical representation of a random situation.

Outcome — One possible result.

Event — An outcome or collection of outcomes being investigated.

Sample space — The complete set of possible outcomes.

Favourable outcome — An outcome that satisfies the event being investigated.

Equally likely — Having the same probability of occurring.

Fair — Describing a probability device in which equivalent outcomes have equal probabilities.

Random — Describing a process whose individual outcome cannot be predicted with certainty.

Impossible event — An event with probability 0.

Certain event — An event with probability 1.

Complement — The event that the original event does not occur.

Complementary events — An event and its complement; their probabilities total 1.

Expected frequency — The number of times an event is predicted to occur based on theoretical probability.

Trial — One repetition of a probability experiment.

Relative frequency — The proportion of trials in which an event occurs.

Random variation — Natural differences between observed results caused by randomness.

Assumption — A condition accepted as true when constructing or using a mathematical model.

Replacement — Returning a selected object before another selection so the original probabilities are restored.

Without replacement — Selecting objects without returning them, causing probabilities to change.

Bias — A systematic influence that favours certain outcomes.

Simulation — A model used to imitate a random process.

Probability tree — A diagram used to organize outcomes and probabilities across several stages.

Expected outcome — A result anticipated from a theoretical model.

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Key Takeaways

• Theoretical probability is calculated using a mathematical model rather than observed experimental results.
• For equally likely outcomes:

P(event) = favourable outcomes / total possible outcomes

• The sample space contains all possible outcomes.
• Favourable outcomes are the outcomes that satisfy the event being investigated.
• The simple counting formula assumes that the individual outcomes are equally likely.
• Fair coins, fair dice, and equal spinner sections are common examples of equally likely outcomes.
• An impossible event has probability 0.
• A certain event has probability 1.
• Probabilities can be expressed as fractions, decimals, or percentages.
• Probabilities must always lie between 0 and 1.
• Complementary events have probabilities that total 1.
• The complement rule is:

P(not A) = 1 − P(A)

• Theoretical probability can be used to calculate expected frequencies.
• Expected frequency is:

theoretical probability × number of trials

• Expected frequency is not a guaranteed result.
• Theoretical probability and experimental probability may differ.
• Random variation causes experimental results to fluctuate.
• With many trials, experimental probability often becomes closer to theoretical probability.
• Theoretical probability depends on assumptions.
• Common assumptions include fairness, equal likelihood, random selection, and consistent experimental conditions.
• If those assumptions are incorrect, the theoretical model may not accurately represent reality.
• Strong and persistent differences between theoretical and experimental results may suggest that the model or experiment should be investigated.
• Replacement can keep probabilities constant between repeated selections.
• Without replacement, probabilities may change after each selection.
• Probability trees and systematic lists help organize multi-stage experiments.
• Probability models are simplifications of real situations and should always be interpreted with their assumptions in mind.

When solving a theoretical probability problem, remember:

Outcomes → Assumptions → Favourable Outcomes → Calculate → Check → Interpret

Always ask:

What are all the possible outcomes?

Are they equally likely?

Which outcomes satisfy the event?

What assumptions am I making?

Does my probability fall between 0 and 1?

How would I expect experimental results to compare with this theoretical value?

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Check Your Understanding

1. Define theoretical probability.

2. Write the formula for theoretical probability when outcomes are equally likely.

3. What is a sample space?

4. Write the sample space for a standard six-sided die.

5. Calculate the probability of rolling a 5.

6. Calculate the probability of rolling an even number.

7. Calculate the probability of rolling a number greater than 2.

8. Calculate the probability of rolling a 9.

9. Calculate the probability of rolling a number less than 7.

10. A bag contains 5 red, 4 blue, and 1 green counter. Calculate:

a. P(red)
b. P(blue)
c. P(green)
d. P(not red)

11. Which colour in Question 10 is most likely to be selected?

12. A spinner has eight equal sections: three red, two blue, two green, and one yellow. Calculate the theoretical probability of each colour.

13. Which colours in Question 12 are equally likely?

14. Explain why simply counting the number of colours would give incorrect probabilities if the spinner sections were different sizes.

15. A standard deck contains 52 cards. Calculate:

a. P
b. P(red card)
c. P(ace)

16. A fair coin is flipped 500 times. How many heads would you expect?

17. A fair die is rolled 600 times. How many sixes would you expect?

18. Explain why the actual number in Question 17 might not be exactly equal to the expected number.

19. A fair die is rolled 120 times and produces 24 sixes. Calculate the experimental probability of rolling a 6 and compare it with the theoretical probability.

20. Does the result in Question 19 prove that the die is unfair? Explain.

21. What assumptions are made when we say that P(heads) = 1/2 for a coin?

22. Explain why a weighted die may not satisfy the usual theoretical probability model.

23. A bag contains equal-sized counters, but a student looks inside before selecting one. Which assumption of the theoretical model has been violated?

24. A bag contains 5 red and 5 blue counters. One red counter is selected and not replaced. What is the probability of selecting red on the next draw?

25. Explain why probabilities can change when objects are selected without replacement.

26. A fair coin is flipped twice. List all possible outcomes.

27. Using your answer to Question 26, calculate:

a. P(two heads)
b. P(exactly one head)
c. P(no heads)

28. A coin is flipped and a six-sided die is rolled. How many combined outcomes are possible?

29. Calculate the probability of obtaining tails and an odd number in Question 28.

30. Challenge: A game uses a spinner divided into 12 equal sections. Five sections are red, four are blue, two are green, and one is yellow.

a. Calculate the theoretical probability of each colour.
b. Predict the expected frequency of each colour in 600 spins.
c. An experiment produces 261 red, 188 blue, 101 green, and 50 yellow results. Calculate the experimental probability of each colour.
d. Compare the theoretical and experimental probabilities.
e. Explain why the results are not expected to match the theoretical probabilities exactly.
f. Identify the assumptions that must be true for the theoretical model to be appropriate.