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D2.5 · Investigate long-run convergence of experimental probability

Learn to investigate long-run convergence of experimental probability through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Data Management

Investigating long-run convergence of experimental probability | MBF3C study topic D2.5

Suppose you toss a coin a few times and get heads more often than tails. Does that mean the coin is unfair? Not necessarily. With only a few tosses, results can vary quite a bit. If you keep tossing the coin, the proportion of heads may settle closer to one-half. In this lesson, you will investigate this pattern, called long-run convergence of experimental probability. Convergence means that a value tends to settle near a number as more data are collected. It does not mean every new result is predictable or that the value must become exact.

What you will learn

Explain experimental probability as a fraction based on observed results.
Investigate how experimental probability can change as the number of trials increases.
Describe long-run convergence without claiming that results become exactly equal to theoretical probability.
Use a table or repeated-trial model to compare experimental probabilities at different points.

1. A probability bridge: count and compare

A probability describes how likely an outcome is. For an experiment, experimental probability uses results that have actually been observed. A trial is one run of the experiment, such as one coin toss. An outcome is a possible result, such as heads. The number of times an outcome occurs is its frequency.

To find experimental probability, divide the frequency of the outcome by the total number of trials. For example, if a spinner lands on blue 7 times in 20 spins, the experimental probability of blue is

\frac{7}{20}

, or

0.35

. This is an observed proportion, not a promise about the next spin.

A proportion is a part compared with a whole. Grade 10 skills with fractions, decimals, and percentages are useful here: the same proportion can be written in different forms. When results are added, both the frequency and the total number of trials can change, so the experimental probability can change too.

For an outcome

A

, let

f_A

represent its observed frequency and let

N

represent the total number of trials. Divide the frequency by the total to find the experimental probability.

P_{\mathrm{exp}}(A)=\frac{f_A}{N}

Experimental probability is calculated from observed results.
A larger number of trials means more observations are included, but it does not guarantee a particular outcome on the next trial.

2. What long-run convergence means

Imagine recording the proportion of heads after each group of coin tosses. Early proportions may jump around: a short run might have many heads, and another short run might have few. As the number of tosses grows, the overall proportion often settles into a narrower range near one-half for a fair coin. This pattern is long-run convergence.

The word “often” matters. Convergence is a pattern to investigate, not a guarantee that the proportion will move closer after every toss. A run can move away from one-half when new results are added. Also, a proportion based on a finite number of trials may not equal one-half exactly, even when the coin is fair.

For a fair coin, one-half is the theoretical probability of heads: it comes from the model that each side is equally likely. Experimental probability is the value calculated from results. Comparing the two helps you see how observed results behave as the number of trials increases. The goal is to notice the long-run pattern, not to claim that a short run proves a coin is fair or unfair.

P_{\mathrm{theoretical}}(\text{heads})=\frac{1}{2}

Short runs can vary noticeably.
With many trials, experimental probability may settle near the theoretical probability for a fair model.
The experimental probability can still move up or down and need not become exact.

3. Investigate with a running record

A running record shows results at growing totals, such as after 10, 50, and 100 trials. For each total, calculate the outcome frequency divided by that total. Comparing these values makes change visible. A graph can also help: put the number of trials along the horizontal axis and the experimental probability along the vertical axis. A horizontal reference at the theoretical probability can help you compare the observed pattern with the model.

You can investigate with physical trials or a random-number tool. For a coin model, use two equally likely results, one for heads and one for tails. Record every trial rather than choosing only the results that fit your expectation. If you use a simulation, repeat it with a different run as well. Two runs will not usually have identical values, but both may show early variation and later settling near one-half.

When interpreting a table or graph, ask: What is the total number of trials at each point? What is the experimental probability? How much did it change between recorded points? Is it generally near the theoretical value in the longer run? These questions support an evidence-based description. They do not require a fixed distance from the theoretical value or a prediction of the exact next result.

P_{\mathrm{exp}}(\text{heads after }n\text{ trials})=\frac{H_n}{n}

Record all trials and state the total used for each calculation.
Compare several running totals rather than relying on one short run.
Describe the pattern shown by the data; do not present one simulated run as a universal result.

Illustrative cumulative coin-toss record

Total tosses	Heads so far	Experimental probability of heads
10	7	0.70
50	28	0.56
200	103	0.515

Worked example

Example 1: A coin-toss record

A student tosses a coin 20 times and records 13 heads. After 80 tosses in total, the record shows 43 heads. Calculate the experimental probability of heads at both points and describe the change.

Calculate the first proportion
At 20 tosses, divide the 13 observed heads by the 20 total tosses. This gives the fraction of recorded trials that were heads.
$\frac{13}{20}=0.65$
Calculate the later proportion
At 80 tosses, use the total number of heads recorded by that point, 43, and divide by all 80 tosses. The later total includes the first 20 tosses.
$\frac{43}{80}=0.5375$
Compare with the model
The first proportion is $0.65$ , while the later one is $0.5375$ . The later value is closer to $0.5$ , the theoretical probability for a fair coin. This record is consistent with settling nearer one-half, but it does not prove that every longer run will do so.
$0.65>0.5375>0.5$

Answer: The experimental probabilities are 0.65 after 20 tosses and 0.5375 after 80 tosses. In this record, the proportion moved closer to one-half.

Check: The frequencies are possible: 13 heads in the first 20, and 43 heads in 80 overall. The proportion after 80 trials is based on the cumulative total, not just the later 60 tosses.

Worked example

Example 2: A simulated spinner

A computer simulates a spinner with two equally likely outcomes, green and yellow. After 10 spins, green has appeared 3 times. After 100 spins, green has appeared 46 times. Compare the experimental probabilities and explain what this run suggests.

Find the early proportion
Use the 3 green results out of the first 10 spins. Expressing the fraction as a decimal makes it easy to compare with the later value.
$\frac{3}{10}=0.30$
Find the later proportion
At 100 spins, green appeared 46 times in total. Divide that frequency by the total number of spins to find the cumulative experimental probability.
$\frac{46}{100}=0.46$
Interpret the evidence
The model gives green a theoretical probability of one-half. The early value, $0.30$ , is farther from one-half than the later value, $0.46$ . This run shows the proportion settling nearer one-half as the trial count grows, though a different simulation could show a different path.
$|0.30-0.50|=0.20,\quad |0.46-0.50|=0.04$

Answer: The experimental probabilities are 0.30 after 10 spins and 0.46 after 100 spins. In this simulation, the later value is closer to the theoretical probability of 0.50.

Check: The differences from 0.50 are 0.20 and 0.04, respectively. This supports the comparison, but one simulation does not establish what every run will do.

Common mistakes and how to avoid them

Assuming the experimental probability must get closer to the theoretical probability after every trial.

Correction: New outcomes can move the proportion either way. Look for the overall pattern across increasing totals, not a steady step-by-step movement.

Using only the most recent group of trials when asked for the probability after a total number of trials.

Correction: For a cumulative probability, use the outcome frequency and total from the full run up to that point.

Saying that a long run must equal the theoretical probability exactly.

Correction: Say that the experimental probability may settle near the theoretical probability. A finite run can still differ.

Treating one unusual short run as proof that the model is wrong.

Correction: Short runs can vary. Gather more trials and compare the longer-run pattern before making a careful statement.

Lesson summary

Experimental probability is an observed frequency divided by the number of trials.
Long-run convergence describes an experimental probability settling near a theoretical probability as trials accumulate.
The value can fluctuate, and it is not required to equal the theoretical probability exactly.
Use cumulative records, calculations, and clear descriptions to investigate the pattern.

Check your understanding

Question 1

A fair coin has 18 heads in 30 tosses. What is the experimental probability of heads?

0.40
0.50
0.60
0.80

Show answer and explanation

0.60

Divide the 18 heads by 30 total tosses:

\frac{18}{30}=0.60

Question 2

A model predicts a theoretical probability of 0.50. A running record gives 0.42 after 25 trials and 0.48 after 200 trials. Which statement is best supported?

The experimental probability increased and the later value is closer to 0.50.
The experimental probability must now remain at 0.48.
The model is incorrect because neither value is exactly 0.50.
The next trial must produce the less common outcome.

Show answer and explanation

The experimental probability increased and the later value is closer to 0.50.

The later value is closer to 0.50 than the earlier value. These results show one pattern; they do not guarantee what future values or outcomes will be.

Question 3

Why can a short run differ quite a bit from the theoretical probability?

A short run includes only a limited number of observed trials.
Theoretical probability changes after every trial.
Experimental probability always excludes repeated outcomes.
A short run determines the exact result of a longer run.

Show answer and explanation

A short run includes only a limited number of observed trials.

With few trials, each result has a larger effect on the observed proportion. A short run does not determine the results of a longer run.

Key terms

Trial: One run of an experiment, such as one spin or one toss.
Outcome: A possible result of an experiment, such as heads on a coin toss.
Frequency: The number of times an outcome appears in recorded trials.
Experimental probability: The observed frequency of an outcome divided by the total number of trials.
Theoretical probability: A probability predicted by a model of the experiment, such as one-half for either side of a fair coin.
Long-run convergence: The pattern in which an experimental probability tends to settle near a theoretical probability as more trials are included.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D2.5. It is a study resource, not an official curriculum publication.

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