D2.4 Compare theoretical and experimental probability

Ontario Grade 11 Mathematics

Data Management

MBF3C study topic D2.4

Probability describes how likely an event is. An event is an outcome, or group of outcomes, that we are interested in. For example, rolling an even number on a number cube is an event. It includes the outcomes 2, 4, and 6. Theoretical probability comes from the possible outcomes in a model. Experimental probability comes from what happens in trials. Comparing them helps us describe how observed results relate to the model’s prediction.

What you will learn

Explain the difference between theoretical and experimental probability.
Calculate both probabilities for situations with equally likely outcomes.
Compare a model’s prediction with results observed in trials.
Describe what a comparison says about the data.

1. Prerequisite bridge: outcomes, events, and trials

An outcome is one possible result of an activity. A number cube has six possible outcomes: 1, 2, 3, 4, 5, and 6. An event is the result, or group of results, that we are checking. For example, rolling an even number is an event with three outcomes: 2, 4, and 6.

A trial is one attempt at an activity. One roll of a number cube is one trial. If you roll it 20 times, you have 20 trials. The total number of trials tells you how many results were observed.

A fraction compares a part with a whole. Probability uses a fraction to compare outcomes that fit an event with a relevant total. When outcomes are equally likely, each has the same chance of occurring. For example, a fair number cube is modelled as having six equally likely outcomes.

A probability can be written as a fraction, decimal, or percent. These are different ways to express the same value. For instance,

\frac{1}{2}

is equal to 0.5 and 50%. A probability of 0 means an event cannot occur in the model; a probability of 1 means it is certain in the model.

0\leq P(E)\leq1

An event can include one possible outcome or several.
A trial is one attempt; many trials produce a set of observed results.
For equally likely outcomes, compare the number that fit the event with the total number of outcomes.

2. Theoretical probability: what the model predicts

Theoretical probability is calculated from a model of the situation. You do not need to perform the activity to find it. First, identify the possible outcomes. Then count the outcomes that fit the event and compare that count with the total number of equally likely outcomes. This method depends on the outcomes being equally likely.

For a fair number cube, the possible outcomes are 1 through 6. Three of the six outcomes are even: 2, 4, and 6. The model’s probability of an even result is therefore

\frac{3}{6}

, which is equal to

\frac{1}{2}

or 0.5.

This probability is a prediction based on the model. It does not promise that every group of six rolls will include exactly three even results. A small number of trials may give more even results or fewer even results than the model’s probability suggests.

The model’s assumptions matter. If a spinner is described as fair and has equal-sized sections, its sections can be treated as equally likely. If you do not know whether the sections are equal or the spinner is fair, you should not assume that its outcomes are equally likely.

For an event

E

, theoretical probability is the number of equally likely outcomes in

E

divided by the total number of equally likely outcomes.

P_{\mathrm{theory}}(E)=\frac{\text{favourable outcomes}}{\text{equally likely outcomes}}

Theoretical probability comes from possible outcomes in a model.
Use the equally likely outcome method only when its assumption fits the situation.
A prediction does not guarantee an exact result in a small set of trials.

3. Experimental probability: what the trials show

Experimental probability is calculated from observed results. Count how many trials produced the event, then compare that count with the total number of trials. This value summarizes the data that were collected.

Suppose a number cube is rolled 20 times and an even number appears 13 times. The experimental probability of an even result is

\frac{13}{20}

, or 0.65. This is greater than the theoretical probability of 0.5. The values can both be correct: one describes the model, while the other describes those particular rolls.

When comparing values, name which one is theoretical and which one is experimental. State whether the experimental value is greater than, less than, or equal to the theoretical value. If you describe the difference, use the same form for both probabilities. In this example, the difference is

0.65-0.50=0.15

.

The observed count must remain as recorded. Experimental probability describes what happened, not what you think should have happened. A difference from the theoretical probability does not by itself mean that either calculation is wrong.

P_{\mathrm{exp}}(E)=\frac{\text{times }E\text{ occurred}}{\text{total trials}}

Experimental probability uses observed counts from trials.
The denominator is the total number of trials.
A difference between the two values is possible.

4. Making a useful comparison

A fair comparison is about the same event in both cases. For example, compare the theoretical and experimental probabilities of rolling an even number. Do not compare the theoretical probability of rolling an even number with the experimental probability of rolling a 6; those are different events.

It also helps to write both values in the same form, such as decimals. Then it is easier to tell which is greater. Include the event, the two values, and a short statement about how they relate.

More trials give you more observed results to compare with the model, but they do not guarantee that experimental probability will equal theoretical probability. Report the values calculated from the data. Do not change the data to make them match the prediction, and do not claim that a difference alone proves a model is fair or unfair.

Compare the same event in both calculations.
Use a matching form, such as decimals, to compare values clearly.
Describe the results without treating the prediction as a required outcome.

Two ways to describe the same event

Probability type	What it uses	Question it answers
Theoretical	Possible outcomes in a model	What probability does the model predict?
Experimental	Observed results in trials	What fraction of the trials showed the event?

Worked example

A number cube: prediction and observed rolls

A fair six-sided number cube is rolled 30 times. It shows a 5 on 7 rolls. Compare the theoretical and experimental probabilities of rolling a 5.

Find the model probability
There is one outcome that is a 5 among six equally likely outcomes. The theoretical probability is the count of matching outcomes compared with all possible outcomes.
$P_{\mathrm{theory}}(5)=\frac{1}{6}\approx0.167$
Calculate from the trials
The 5 appeared on 7 of the 30 rolls. Use 7 as the number of times the event occurred and 30 as the total number of trials.
$P_{\mathrm{exp}}(5)=\frac{7}{30}\approx0.233$
Compare the probabilities
Both values describe rolling a 5. The experimental value is greater. Subtracting the rounded values gives an approximate difference.
$0.233-0.167=0.066$

Answer: The theoretical probability is

\frac{1}{6}

, or about 0.167. The experimental probability is

\frac{7}{30}

, or about 0.233. In these 30 rolls, the experimental probability is greater by about 0.066.

Check: The model has one outcome that is a 5 out of six equally likely outcomes. The trials recorded 7 fives out of 30 rolls. Both calculations concern the same event.

Worked example

A bag of counters: model and trial results

A bag contains 4 red counters and 6 blue counters. One counter is drawn, its colour is recorded, and it is returned before the next draw. In 40 draws, a red counter appears 18 times. Compare the theoretical and experimental probabilities of drawing red.

Count the model outcomes
There are 10 counters in total, and 4 are red. Since each counter is equally likely to be drawn, compare the red counters with all the counters.
$P_{\mathrm{theory}}(\text{red})=\frac{4}{10}=0.4$
Calculate from the recorded draws
Red appeared 18 times in 40 draws. Use the observed count and total number of draws to find the experimental probability.
$P_{\mathrm{exp}}(\text{red})=\frac{18}{40}=0.45$
State the comparison
The experimental probability is greater than the theoretical probability. The subtraction describes the difference between the values; it does not change the model or the recorded results.
$0.45-0.40=0.05$

Answer: The theoretical probability of drawing red is 0.4, or 40%. The experimental probability is 0.45, or 45%. The experimental value is 0.05, or 5 percentage points, greater.

Check: There are 4 red counters among 10 counters, so the model probability is 0.4. The recorded results show 18 red draws among 40, so the experimental probability is 0.45. Both concern drawing red.

Common mistakes and how to avoid them

Using the number of trials as the denominator for theoretical probability.

Correction: Theoretical probability uses the total number of equally likely outcomes in the model. The number of trials is used for experimental probability.

Comparing probabilities for different events.

Correction: Check that both values describe exactly the same event before comparing them.

Assuming experimental probability must match the theoretical probability.

Correction: Theoretical probability comes from the model, while experimental probability reports the trials. Their values can differ.

Changing observed counts because they do not match the prediction.

Correction: Keep the recorded data unchanged. Calculate from what happened, then explain the comparison.

Lesson summary

Theoretical probability is calculated from equally likely outcomes in a model.
Experimental probability is calculated from results observed in trials.
Compare the same event and use matching forms, such as decimals.
The values may differ; describe what the data show.

Check your understanding

Question 1

A fair coin is flipped 20 times and lands heads 12 times. Which statement correctly compares the probabilities of heads?

Theoretical probability is $\frac{1}{2}$ ; experimental probability is $\frac{12}{20}$ , so the experimental value is greater.
Theoretical probability is $\frac{12}{20}$ ; experimental probability is $\frac{1}{2}$ , so the theoretical value is greater.
Both probabilities are $\frac{1}{2}$ because the coin is fair.
Theoretical probability is $\frac{1}{20}$ ; experimental probability is $\frac{12}{20}$ .

Show answer and explanation

Theoretical probability is

\frac{1}{2}

; experimental probability is

\frac{12}{20}

, so the experimental value is greater.

A fair coin has two equally likely outcomes, so the theoretical probability of heads is

\frac{1}{2}=0.5

. The observed fraction is

\frac{12}{20}=0.6

, which is greater.

Question 2

A spinner has 8 equal sections, and 3 are green. In 24 spins, it lands on green 6 times. What is the comparison?

Theoretical probability is $\frac{3}{8}$ ; experimental probability is $\frac{6}{24}$ , so the experimental value is smaller.
Theoretical probability is $\frac{6}{24}$ ; experimental probability is $\frac{3}{8}$ , so the theoretical value is smaller.
Both probabilities are $\frac{3}{8}$ because the spinner has equal sections.
Theoretical probability is $\frac{3}{24}$ ; experimental probability is $\frac{6}{8}$ .

Show answer and explanation

Theoretical probability is

\frac{3}{8}

; experimental probability is

\frac{6}{24}

, so the experimental value is smaller.

The model gives

\frac{3}{8}=0.375

. The trials give

\frac{6}{24}=0.25

. Therefore, the experimental value is smaller.

Question 3

Why can experimental probability differ from theoretical probability?

A limited set of trials can produce a different mix of results from the model’s prediction.
Theoretical probability is calculated using observed trials.
Experimental probability must always be greater than theoretical probability.
A difference means one value was automatically calculated incorrectly.

Show answer and explanation

A limited set of trials can produce a different mix of results from the model’s prediction.

Experimental probability describes results collected in trials. Those results do not have to match the model’s prediction exactly.

Key terms

Outcome: One possible result of an activity.
Event: The outcome or group of outcomes being considered.
Trial: One attempt at an activity, such as one roll or one spin.
Theoretical probability: A probability calculated from possible outcomes in a model.
Experimental probability: A probability calculated from results observed in trials.
Equally likely: Describes outcomes that have the same chance of occurring.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D2.4. It is a study resource, not an official curriculum publication.

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D2.4 · Compare theoretical and experimental probability

Ontario Grade 11 Mathematics

MBF3C study topic D2.4

What you will learn

1. Prerequisite bridge: outcomes, events, and trials

2. Theoretical probability: what the model predicts

3. Experimental probability: what the trials show

4. Making a useful comparison

Two ways to describe the same event

A number cube: prediction and observed rolls

A bag of counters: model and trial results

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MBF3C

About this lesson