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D2.2 · Compare theoretical and experimental probability of independent events

Learn to compare theoretical and experimental probability of independent events through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Data

Use possible outcomes and trial results to study independent events

Imagine tossing a coin. Before the toss, you can think about the coin’s two sides and predict how likely heads is. After several tosses, you can count how often heads actually appeared. These are two ways to describe probability, or chance. One attempt is called a trial. An event is a result, or group of results, that we are thinking about. Events are independent when one event does not change the chance of the other. A coin toss does not change the coin, so it does not change the chance on the next toss. In this lesson, you will compare a chance based on possible outcomes with a chance based on trial results.

What you will learn

Explain the difference between theoretical and experimental probability.
Recognize independent events by checking whether one event changes the chance of another.
Read a probability tree and use it to compare chances.
Compare a prediction with results from a stated set of trials.

Theoretical and experimental probability

A fraction compares a part with a whole. Its top number tells how many parts we count. Its bottom number tells how many parts there are altogether. For a fair coin, heads and tails are equally likely. That means each side has the same chance.

Theoretical probability is a chance found by looking at possible outcomes. An outcome is one result that can happen. To find theoretical probability, count the wanted outcomes and compare that number with all the equally likely outcomes. Equally likely outcomes have the same chance.

For example, a fair coin has two equally likely outcomes: heads and tails. One of the two outcomes is heads, so the theoretical probability of heads is one out of two.

Experimental probability is a chance found from trial results. Count how often the event happened, then compare that count with the total number of trials. If a coin lands heads seven times in twenty tosses, the experimental probability of heads is seven out of twenty.

The two probabilities can be different. Theoretical probability describes a prediction based on possible outcomes. Experimental probability describes what happened in the trials. A small group of trials may not match the prediction exactly.

P(E)=\frac{\text{wanted outcomes}}{\text{equally likely outcomes}}

Theoretical probability uses possible equally likely outcomes.
Experimental probability uses recorded trial results.
Both compare a wanted part with a whole.

Independent events and probability trees

Two events are independent when the first event does not change the chance of the second. With a fair coin, the first toss does not change the coin. Heads and tails remain equally likely on the next toss.

A probability tree is a picture with branches that shows possible results in order. For two coin tosses, the first split shows heads or tails. Each branch then splits into heads or tails for the second toss. At each split, the branch probabilities add to one because the branches show all the possible results at that point.

The four complete routes are heads then heads, heads then tails, tails then heads, and tails then tails. Each route is equally likely in this fair-coin example. The tree helps keep track of results and their order.

A tree does not make events independent. The situation determines whether the chance changes. If a counter is put back into a bag before another draw, the bag has the same contents for the next draw. If it is not put back, the contents change. Check what happens between events before deciding whether they are independent.

\frac{1}{2}+\frac{1}{2}=1

Independent events do not change one another’s chances.
At each split, the probabilities of all branches add to one.
Check what happens between events before deciding whether they are independent.

Compare a prediction with trial results

To compare the two kinds of probability, first name the event. Then find the theoretical probability from possible outcomes and the experimental probability from recorded results.

For a fair coin, one of the two equally likely sides is heads. So the theoretical probability of heads is one-half. This describes the chance on each toss. It does not promise exactly ten heads in every group of twenty tosses.

Suppose a clearly hypothetical set of twenty tosses has seven heads. The experimental probability is seven-twentieths. This is less than one-half. Heads appeared less often in these trials than the theoretical probability describes as a share of tosses.

This comparison tells us about this set of trials. It does not change the chance on a new toss. For independent events, earlier results do not change the chance of the next event.

\frac{7}{20}<\frac{1}{2}

Use possible outcomes to find theoretical probability.
Use recorded trial results to find experimental probability.
A difference between the two does not automatically mean something went wrong.

A clear method

First, name the event you are studying. Decide whether the question asks for a prediction or gives results from trials.

For theoretical probability, list or show the possible outcomes. Count the outcomes that match the event. Compare that count with all equally likely outcomes. A probability tree can help when results happen in order.

For experimental probability, count how many times the event happened. Compare that number with the total number of trials. Use the stated results; do not replace them with a prediction.

If there is more than one event, check whether an earlier result changes the later chance. Then compare the probabilities and explain what the comparison means.

Name the event.
Use possible outcomes for theoretical probability and recorded results for experimental probability.
Check whether the events are independent.
Explain what the comparison says about the stated situation.

Worked example

Heads on both tosses

A fair coin is tossed twice. What is the theoretical probability of getting heads on both tosses?

Complete routes for two coin tosses

Each complete route has probability one-quarter. Together, the four routes have probability one.

Check the tosses
The first toss does not change the coin. The second toss still has the same chance of heads or tails, so the tosses are independent.
Count the possible routes
The four equally likely routes are heads-heads, heads-tails, tails-heads, and tails-tails. Only heads-heads gives heads on both tosses. There is one wanted route out of four equally likely routes.
$\frac{1}{4}$
State the probability
The theoretical probability of heads on both tosses is one out of four.
$P(\text{heads on both})=\frac{1}{4}$

Answer: The theoretical probability is one-quarter.

Check: Only one of the four equally likely routes gives heads on both tosses.

Worked example

Compare a prediction with stated results

In a hypothetical activity, a fair coin is tossed twenty times and lands heads seven times. Compare the theoretical and experimental probabilities of heads.

Find the theoretical probability
A fair coin has two equally likely sides, and one side is heads. Compare the one heads side with both possible sides.
$\frac{1}{2}$
Find the experimental probability
The hypothetical results show seven heads in twenty tosses. Compare the seven heads with all twenty tosses.
$\frac{7}{20}$
Compare the probabilities
Seven-twentieths is less than one-half. In these hypothetical trials, heads appeared less often than the theoretical probability describes as a share of tosses. The results do not change the chance on a new toss.
$\frac{7}{20}<\frac{1}{2}$

Answer: The theoretical probability is one-half. The experimental probability is seven-twentieths, which is less than one-half.

Check: The results are explicitly hypothetical, not real observations.

Worked example

Heads, then tails

A fair coin is tossed twice. What is the theoretical probability of getting heads on the first toss and tails on the second toss?

Complete routes for two coin tosses

Each complete route has probability one-quarter. Together, the four routes have probability one.

Check whether the tosses are independent
The first toss does not change the coin. The second toss still has the same chance of heads or tails. The events are independent.
Find the matching route
The four equally likely routes are heads-heads, heads-tails, tails-heads, and tails-tails. Only heads-tails has heads first and tails second.
$\frac{1}{4}$
State the probability
Theoretical probability compares the one matching route with all four equally likely routes. The probability is one-quarter. P(heads, then tails)= $\frac{1}{4}$

Answer: The theoretical probability is one-quarter.

Check: Order matters: tails-heads does not match heads first and tails second.

Common mistakes and how to avoid them

Thinking a fair coin must land heads exactly half the time in every group of tosses.

Correction: One-half is the theoretical chance on each toss. A group of results may differ from that prediction.

Using trial results as though they were theoretical probability.

Correction: Theoretical probability comes from possible outcomes. Experimental probability comes from results recorded in trials.

Thinking an earlier result changes the next result for independent events.

Correction: For independent events, the first result does not change the next event’s chance. Check whether the situation changes between events.

Treating every route in a probability tree as equally likely without checking.

Correction: Routes are equally likely in these fair-coin examples because every branch has the same chance. Check the branch chances before comparing routes.

Lesson summary

Theoretical probability is based on possible equally likely outcomes.
Experimental probability is based on results from trials.
Independent events do not change one another’s chances.
A probability tree shows possible results in order. At each split, branch probabilities add to one.
Theoretical and experimental probabilities can differ.

Check your understanding

Question 1

A fair coin is tossed twice. What is the theoretical probability of tails on both tosses?

One-quarter
One-half
Three-quarters
One

Show answer and explanation

One-quarter

There are four equally likely routes, and only tails-tails gives tails on both tosses.

Question 2

A hypothetical set of twelve fair-coin tosses has five heads. What is the experimental probability of heads?

One-half
Five-twelfths
Seven-twelfths
Five-halves

Show answer and explanation

Five-twelfths

Experimental probability compares the five heads with all twelve tosses, giving five-twelfths.

Question 3

A counter is drawn from a bag and put back before another draw. Why are the two draws independent?

The bag has fewer counters after the first draw.
The first result changes the chance of the second result.
The bag has the same contents for the second draw, so its chances stay the same.
The second draw must match the first draw.

Show answer and explanation

The bag has the same contents for the second draw, so its chances stay the same.

Putting the counter back restores the same contents, so the chance on the second draw does not change.

Key terms

Event: An outcome or group of outcomes being considered, such as getting heads.
Theoretical probability: A chance found from the possible outcomes in a situation.
Experimental probability: A chance found from results recorded in trials.
Trial: One attempt, such as one coin toss.
Independent events: Events where the result of one does not change the chance of the other.
Probability tree: A picture that uses branches to show possible results in order.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation D2.2. It is a study resource, not an official curriculum publication.

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