D2.3 Run experiments and determine experimental probability

Ontario Grade 11 Mathematics

Data Management

Ontario Grade 11 MBF3C | Study-guide label: D2.3

Probability describes how likely an event is. Experimental probability uses results that you collect by running an experiment. For example, you can toss a coin many times and record how often it lands heads. The estimate comes from what happened in those trials. A different run may give a different result. In this lesson, you will plan a clear experiment, keep track of its results, and use the results to find experimental probability.

What you will learn

Describe how to run an experiment and record its results.
Identify the total number of trials and the number of times an event occurs.
Calculate experimental probability from observed results.
Use an experimental probability as an estimate, not a guarantee.

1. Bridge: Outcomes, events, and trials

An outcome is one possible result of an action. When you roll a standard six-sided number cube, the possible outcomes are the numbers

1

through

6

. An event is the result, or group of results, that you are interested in counting. Rolling an even number is an event because it includes the outcomes

2

,

4

, and

6

.

A trial is one performance of an experiment. One roll of a number cube is one trial. If you roll it

10

times, you have completed

10

trials. Before you begin, state the event clearly. That way, you know exactly which results to count.

You also need a total trial count. This is the number of times you performed the experiment, including trials when the event did not occur. For example, if a coin lands tails, that toss still counts as a trial in an experiment about heads.

An outcome is one possible result.
An event is the result or group of results you are counting.
A trial is one performance of the experiment.

2. Plan and run a clear experiment

An experiment is a planned action used to collect results. Choose a procedure you can repeat in the same way. For a coin-toss experiment, use the same coin and toss it in a consistent way. Decide in advance that the event is, for example, “lands heads.”

Record every trial, not only the results that match what you expect. A tally, a list, or a table can help you keep track. After each trial, record its outcome. At the end, count the total trials and count how many times the event occurred. These two counts are the information you need to calculate experimental probability.

You can run an experiment with physical objects or use a device or simulation. In either case, record how many trials were run and how many times the event occurred. For a useful comparison, keep the event and procedure the same across runs.

Results can vary from one run to another. This is normal. An experiment reports what happened in the trials you performed. It does not promise what will happen on the next trial. A larger set of trials gives you more observations, but it still cannot guarantee a particular result in a future run.

Choose and state the event before you start.
Repeat the procedure consistently and record every trial.
Count the full number of trials as well as the event occurrences.

3. Calculate and interpret experimental probability

Experimental probability is an estimate based on observed results. To calculate it, divide the number of times the event occurred by the total number of trials. In the expression, let

E

stand for the event count and let

T

stand for the total trial count.

The event count goes on top because it is the part of the trials that matched the event. The total trial count goes on the bottom because it includes every trial. The result can be written as a fraction, decimal, or percentage. A fraction shows the two counts; a percentage can make comparisons easier.

For example, if an event occurs

7

times in

20

trials, the experimental probability is

\frac{7}{20}

. Dividing gives

0.35

, which is 35%. This means the event occurred in 35% of those trials. It does not mean that the event must occur exactly

35

times in the next

100

trials.

An experimental probability is between

0

and

1

, inclusive, when written as a decimal, or between 0% and 100% when written as a percentage. The event count cannot be greater than the total number of trials. If your result falls outside these limits, check the counts or calculation.

\frac{E}{T}

Use the event count divided by the total trial count.
The event count must be no greater than the total number of trials.
A decimal or percentage describes the observed results; it is not a guarantee.

4. Use an estimate carefully

An experimental probability describes a particular run. It can also be used as an estimate for similar future trials. The word similar matters: the event and the procedure should remain comparable. Results from tossing one coin do not automatically describe a different object or a different procedure.

When you compare two runs, include the number of trials as well as the percentage. For instance, a result of 50% from

4

trials and a result of 50% from

200

trials have the same percentage, but they come from very different numbers of observations. The trial count helps explain the context of each result.

Use careful wording. Say, “The event occurred in 50% of these trials,” or, “This result suggests about 50% in similar trials.” Avoid saying that the event must happen in exactly half of a future set. Experimental probability is an estimate drawn from results, and another run may differ.

Keep the event and procedure comparable when using a result as an estimate.
Report the number of trials when it helps explain the results.
Describe what happened; do not promise what must happen next.

Worked example

A coin-toss experiment

A student tosses a coin

40

times. It lands heads

23

times. Find the experimental probability of heads as a fraction and a percentage.

Identify the event count
The event is “lands heads.” The recorded number of times it occurred is $23$ . The total number of trials is $40$ .
Compare the counts
Put the event count over the total trial count. This gives the fraction of the observed tosses that landed heads.
$\frac{23}{40}$
Convert to a percentage
Divide $23$ by $40$ to get $0.575$ . Multiply the decimal by $100$ to express it as a percentage.
$0.575× 100%=57.5%$

Answer: The experimental probability of heads is

\frac{23}{40}

, or 57.5%.

Check: The event count,

23

, is less than the total of

40

, so the fraction is between

0

and

1

. Also,

23\div 40=0.575

, which is 57.5%.

Worked example

Testing a coloured counter

A student draws one counter from a bag, records its colour, replaces it, and mixes the bag before the next draw. In

60

trials, a blue counter is drawn

18

times. Find the experimental probability of drawing blue. Use the result to estimate how many blue draws might occur in

150

similar trials.

Find the observed probability
Blue is the event. Compare the $18$ blue draws with all $60$ trials. Divide both parts of the fraction by $6$ to simplify it, then write the result as a decimal and percentage.
$\frac{18}{60}=\frac{3}{10}=0.30=30%$
Estimate for a new set of trials
Use the observed fraction as an estimate for similar trials. Find three tenths of $150$ . This gives an estimated count, not a guaranteed result.
$150×\frac{3}{10}=45$

Answer: The experimental probability of drawing blue is

\frac{3}{10}

, or 30%. The result suggests about

45

blue draws in

150

similar trials.

Check: Thirty percent of

150

is

45

. The new set of trials may have more or fewer than

45

blue draws.

Common mistakes and how to avoid them

Putting the total number of trials on top of the fraction.

Correction: Put the event count on top and the total number of trials on the bottom.

Treating an experimental probability as a guarantee for future trials.

Correction: Describe it as an estimate based on the observed results. A future run can give a different result.

Leaving some trials out of the total.

Correction: Record each trial and use the full trial count as the denominator.

Reporting a percentage without saying how many trials were performed.

Correction: Include the trial count when it helps someone understand the results.

Lesson summary

Define the event and plan a repeatable procedure before starting.
Record every outcome and count the total trials and event occurrences.
Calculate experimental probability by dividing the event count by the total trial count.
Use the result as an estimate for similar trials, not as a guarantee.

Check your understanding

Question 1

A number cube is rolled

25

times. The result is

6

on

5

rolls. What is the experimental probability of rolling a

6

?

$\frac{5}{25}=20\%$
$\frac{25}{5}=20\%$
$\frac{6}{25}=24\%$
$\frac{5}{6}\approx83.3\%$

Show answer and explanation

\frac{5}{25}=20\%

The event occurred

5

times out of

25

trials. Thus,

\frac{5}{25}=\frac{1}{5}=0.20=20\%

.

Question 2

A spinner lands on green

14

times in

35

spins. Which statement is supported by this experiment?

Green must occur exactly 40% of the time in every future group of spins.
Green occurred in 40% of these spins, which can be used as an estimate for similar trials.
Green occurred in 35% of these spins.
The experiment had $14$ trials.

Show answer and explanation

Green occurred in 40% of these spins, which can be used as an estimate for similar trials.

The observed fraction is

\frac{14}{35}=\frac{2}{5}=40\%

. It describes these spins and can serve as an estimate, not a guarantee.

Key terms

Outcome: One possible result of an action.
Event: The result or group of results being counted.
Trial: One performance of an experiment.
Experimental probability: An estimate based on the number of times an event occurred and the total number of trials.

Continue through MBF3C

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D2.3. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

D2.3 · Run experiments and determine experimental probability