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Lesson 16 ~30 min Unit 4 · Probability +85 XP

Experimental Probability

Discover how running experiments reveals probability. Learn relative frequency, the law of large numbers, and why more trials always beats fewer.

Today's hook: A coin is flipped 10 times and gets heads 7 times. Does that mean P(heads) = 0.7? No, that's experimental probability, not theoretical. Flip it 10,000 times and you'll get very close to 0.5. This is the law of large numbers in action.
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Think First
warm-up

Before you read on, quickly: A bag holds red and blue marbles. You reach in and pull out a red marble 6 times out of 10 tries. Does that mean exactly 60% of the marbles are red? What would you need to do to be more confident?

Record your answer in your workbook.
1
The Big Idea
+5 XP

Experimental probability (also called relative frequency) is the probability you find by actually running an experiment and recording results. It equals the number of times the event occurred divided by the total number of trials.

Unlike theoretical probability (which you calculate from equally likely outcomes), experimental probability depends on real data from real trials. As you do more and more trials, your experimental probability gets closer and closer to the true theoretical probability. This settling behaviour is called the law of large numbers.

Coin Flip: P(heads) 0.5 trials → P(H) 10 100 1000
Experimental P(event) = frequency ÷ total trials
Count the trials
Total trials is how many times you ran the experiment, not how many outcomes are possible.
More trials = better
The law of large numbers: as trials increase, experimental probability stabilises toward the true value.
Results vary
Small experiments can give unusual results just by chance, that's normal.
2
What You'll Master
objectives

Know

  • The formula: experimental P = frequency ÷ total trials
  • Another name for experimental probability is relative frequency
  • What the law of large numbers states

Understand

  • Why small experiments give unreliable results
  • Why experimental probability approaches theoretical with more trials
  • Why you shouldn't expect experimental results to perfectly match theory

Can Do

  • Calculate experimental probability from a frequency table
  • Compare experimental probability to theoretical probability
  • Explain why results differ and what would improve accuracy
3
Words You Need
vocabulary
Experimental probabilityProbability found by running an experiment: frequency ÷ total trials.
Relative frequencyAnother name for experimental probability. Both mean the same thing.
TrialOne run of an experiment (e.g. one coin flip, one die roll).
FrequencyHow many times an event actually occurred in the experiment.
Law of large numbersAs the number of trials increases, experimental probability gets closer to theoretical probability.
Long-run proportionThe value experimental probability settles towards after many trials, equal to theoretical probability.
4
Spot the Trap
heads-up

Wrong: "I flipped a coin 10 times and got 7 heads, so P(heads) = 0.7 from now on."

Right: That's the experimental probability for that experiment. It doesn't change the true theoretical probability of 0.5. With more trials, it will move closer to 0.5.

Wrong: Expecting experimental results to match theory exactly, even with just a few trials.

Right: Experimental results vary because of random chance. Only with very large numbers of trials do they reliably reflect theoretical probability.

5
Calculating Relative Frequency
+5 XP

To find experimental probability, divide the frequency (how often the event happened) by the total number of trials. You can express this as a fraction, decimal, or percentage.

A die is rolled 50 times. The results are recorded in a table. We can find the relative frequency for each outcome. For example, if 6 appears 9 times, then experimental P(6) = 9 ÷ 50 = 0.18. Compare this to the theoretical value of 1/6 ≈ 0.167.

Outcome Freq Rel. Freq 188/50 = 0.16 21010/50 = 0.20 377/50 = 0.14 499/50 = 0.18 577/50 = 0.14 699/50 = 0.18 Total501.00 All rel. frequencies add to 1
Experimental P(6) = 9 ÷ 50 = 0.18
Check your sum
All relative frequencies for all outcomes should add up to exactly 1 (or 100%).
Simplify fractions
Write 9/50 as a fraction in lowest terms or as a decimal: 0.18.
Denominator = total trials
Never use the number of possible outcomes as the denominator here.
6
The Law of Large Numbers
+5 XP

The law of large numbers tells us that as the number of trials grows, the experimental probability will get closer and closer to the theoretical probability. This is why scientists and researchers use large sample sizes.

Look at this table of coin flip results. With only 10 flips, the experimental probability is 0.70 far from the true 0.5. With 100 flips, it's 0.54. With 1000 flips, it's 0.503 extremely close. The more trials, the more reliable the estimate.

Trials Heads Exp. P 10 7 0.70 100 54 0.54 1 000 503 0.503 10 000 5002 0.5002 Theoretical P(heads) = 0.5 More trials → closer to 0.5
More trials → experimental P → theoretical P
It never guarantees
More trials make results more reliable, but still not a guarantee of exact match.
Variation is normal
Small samples fluctuate a lot. This is expected, not a sign something is wrong.
Used everywhere
Insurance, medicine, and sport all use large sample sizes for this reason.
7
Comparing Experimental and Theoretical
+5 XP

When you compare experimental and theoretical probabilities, differences are expected, especially with small samples. The key question is: how large is the difference, and how many trials were used?

A die is rolled 600 times. Theory says each face should appear 1/6 × 600 = 100 times. Experimental results are close but not exact. A small difference (e.g. 98 vs 100) is normal variation. A large difference (e.g. 50 vs 100) with many trials might suggest the die is biased (not fair).

600 die rolls (expected: 100 each) 100 1 2 3 4 5 6 Small differences = normal variation
Large difference + large n → possibly biased
Calculate expected
Expected frequency = theoretical P × number of trials.
Context matters
A difference of 5 from 100 is small. A difference of 5 from 10 is huge.
Never say "wrong"
Experimental results are never "wrong", they just vary from theoretical due to chance.

Watch Me Solve It · 3 examples

Watch Me Solve It · Find experimental probability
+15 XP per step
Q1
PROBLEM
A die is rolled 50 times. The number 6 appeared 9 times. Find the experimental probability of rolling a 6.
  1. 1
    Identify frequency and total trials
    Frequency of 6 = 9    Total trials = 50
    The frequency is how many times the event happened. The total trials is how many times the experiment was run.
  2. 2
    Apply the formula
    Experimental P(6) = 9 ÷ 50 = 9/50
  3. 3
    Convert to decimal and compare to theory
    9/50 = 0.18    Theoretical P(6) = 1/6 ≈ 0.167
    The experimental probability (0.18) is close to but not equal to the theoretical (0.167). With more trials, they would get closer.
AnswerExperimental P(6) = 9/50 = 0.18. Theoretical is 1/6 ≈ 0.167. Close but not equal, expected with 50 trials.
Watch Me Solve It · Explain the difference
+15 XP per step
Q2
PROBLEM
The die from Q1 was rolled 50 times and 6 appeared 9 times. The theoretical P(6) = 1/6. Explain why the results differ and what would reduce the difference.
  1. 1
    State the experimental vs theoretical values
    Experimental: 0.18    Theoretical: 1/6 ≈ 0.167
    The difference is 0.013, small but present.
  2. 2
    Explain why they differ
    Random variation in 50 trials causes differences by chance
    50 trials is not a large number. Random chance can make some outcomes appear more or less than expected.
  3. 3
    How to reduce the difference
    Use more trials, e.g. 500 or 5000 rolls
    The law of large numbers: as trials increase, experimental probability gets closer to theoretical.
AnswerThe difference is due to random variation in a small sample. To get closer to 1/6, increase the number of trials significantly.
Watch Me Solve It · Plan a simulation
+15 XP per step
Q3
PROBLEM
You want to simulate the probability that a basketball player makes a free throw, given that theoretically they make 1 in 3. How could you use a die to simulate this?
  1. 1
    Identify what probability to model
    P(make) = 1/3, P(miss) = 2/3
    We need a random device where one outcome has probability 1/3.
  2. 2
    Assign die outcomes
    Roll 1 or 2 = "make" (2/6 = 1/3)    Roll 3,4,5,6 = "miss" (4/6 = 2/3)
  3. 3
    Describe how to run it
    Roll the die many times (e.g. 120 times) and count how often 1 or 2 appears
    More rolls = more reliable simulation. Expected: about 40 "makes" in 120 rolls.
AnswerRoll 1 or 2 = make (prob 1/3). Roll 3–6 = miss (prob 2/3). Run 120+ rolls and record relative frequency of "make".
9
Common Pitfalls
heads-up
Using the number of outcomes, not total trials, as the denominator
A die has 6 possible outcomes, but if you rolled it 50 times, the denominator in your relative frequency is 50, not 6. Students often confuse the two.
Fix: Denominator = total number of trials. Numerator = how many times the specific event occurred.
Thinking experimental = theoretical after few trials
Getting 7 heads in 10 flips does NOT mean the coin is biased. With only 10 flips, extreme results happen often just by chance.
Fix: Only draw conclusions about bias after many trials (hundreds or thousands). Small samples give unreliable estimates.
Expecting perfect agreement between experimental and theoretical
Even with 1000 fair coin flips, you might get 512 heads instead of exactly 500. This is normal, probability gives long-run expectations, not guaranteed counts.
Fix: Probability predicts proportions over the long run. Individual experiments will vary. Both 498 and 507 are perfectly normal results for 1000 flips.
Copy Into Your Books

Experimental Probability

  • Also called: relative frequency
  • Formula: P(event) = frequency ÷ total trials
  • Based on real experiment results
  • All relative frequencies add to 1

Law of Large Numbers

  • More trials → more reliable estimate
  • Exp. probability approaches theoretical
  • Small samples have high variation
  • Long-run proportion = theoretical probability

Comparing Results

  • Small differences = normal variation
  • Large diff. + large n = possibly biased
  • Expected freq = P × n
  • Never say results are "wrong"

Key Vocabulary

  • Trial = one run of experiment
  • Frequency = count of successes
  • Simulation = using chance device to model probability
9
Observed Probability
exam wording

This lesson has called it experimental probability. The NSW syllabus and exam papers use a third name for the same idea: observed probability. All three names describe one thing, so treat them as interchangeable.

Observed probability is the relative frequency that results from repeated trials of a simulation. You run the experiment many times, count how often the event actually happened, and divide by the number of trials. Nothing is assumed about fairness; the number comes purely from what was observed.

Observed probability = experimental probability = relative frequency. Observed probability = (number of times the event happened) divided by (number of trials).

Contrast with theoretical probability, which is calculated in advance by assuming every outcome is equally likely. Theoretical says what should happen under fair conditions; observed reports what did happen when you ran the trials.

Observed = what happened divided by trials  |  Theoretical = favourable divided by total, assuming fairness
Read the question's wording
"Observed", "experimental" and "relative frequency" all mean count the trials. "Theoretical" or "expected" means use the formula instead.
More trials, closer agreement
As the number of trials grows, the observed probability tends towards the theoretical probability. That is the law of large numbers from Card 5.
10
Repeating an Experiment with a Random Number Generator
digital tools

Rolling a die 500 times by hand takes an afternoon. A random number generator does it in a moment, which lets you explore relative frequencies over far more trials than you could run physically.

Every calculator and spreadsheet has one. The method is always the same: map each outcome of your experiment to a number, generate as many numbers as you want trials, then count how often each outcome appeared.

Step 1, map the outcomes. A fair die has 6 equally likely outcomes, so generate whole numbers from 1 to 6. A coin has 2, so generate 1 or 2 and let 1 mean heads.

Step 2, generate the trials. On a spreadsheet, =RANDBETWEEN(1,6) filled down 500 rows gives 500 die rolls. On a calculator, the RanInt# or Ran# key does the same one roll at a time.

Step 3, count and divide. If the number 4 appeared 79 times in 500 trials, the observed probability of a 4 is $\tfrac{79}{500} = 0.158$. The theoretical probability is $\tfrac{1}{6} \approx 0.167$, so the two are close.

Map outcomes to numbers, generate many trials, then observed probability = count divided by trials

Wrong: Expecting the generator to give exactly 100 fours in 600 rolls because the theoretical probability is $\tfrac{1}{6}$. Random results vary, and an exact match would be surprising rather than reassuring.

Right: Expecting the observed probability to sit near $\tfrac{1}{6}$, and to sit closer as the number of trials grows. Run 50 trials, then 500, then 5000, and watch the gap shrink.

Wrong: Generating numbers 1 to 5 to simulate a die because you forgot one face. The generator can only be as correct as the map you gave it.

Right: Check that the range of numbers matches the number of equally likely outcomes before generating a single trial.

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Experimental Probability
4 problems

Four drill problems to sharpen your experimental probability skills. Work each, then reveal the answer.

  1. 1 In 80 coin flips, heads appeared 38 times. What is the experimental probability of heads?

    Frequency = 38, total trials = 80. P(heads) = 38/80 = 19/40 = 0.475.P(heads) = 19/40 = 0.475

  2. 2 The theoretical P(heads) for a fair coin is 1/2 = 0.5. Is the experimental result from Q1 surprising? Why or why not?

    Not surprising. 0.475 is very close to 0.5. With 80 trials, getting 38 instead of 40 heads is normal random variation.0.475 vs 0.5, normal variation for 80 trials

  3. 3 Would you trust experimental probability results from just 10 trials? Explain your reasoning.

    No. With only 10 trials, random variation is very high. Results can be far from theoretical probability just by chance. You need many more trials for a reliable estimate.10 trials = very unreliable. Need 100+ for better estimates.

  4. 4 A die is rolled 600 times. About how many times would you expect the number 3 to appear?

    Theoretical P(3) = 1/6. Expected frequency = 1/6 × 600 = 100.Expected frequency = 1/6 × 600 = 100 times
Complete in your workbook.

Quick Check · 6 questions

1
A spinner is spun 40 times. Red appears 12 times. What is the experimental probability of red?
+10 XP
2
What happens to experimental probability as the number of trials increases?
+10 XP
3
Another name for experimental probability is:
+10 XP
4
A fair die is rolled 300 times. How many times would you expect to roll a 5?
+10 XP
5
A fair coin is flipped 10 times and lands heads 7 times. The best explanation is:
+10 XP
6
Two students draw marbles (with replacement) from the same bag to estimate P(red). Student A makes 8 draws and gets 6 red (75%). Student B makes 200 draws and gets 142 red (71%). Whose estimate of P(red) is more reliable?
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Medium 3 MARKS

Q6. A bag contains red and blue marbles. In 60 draws (with replacement), red was drawn 22 times. Find the experimental probability of drawing red. Express it as a fraction, decimal and percentage.

Answer in your workbook.
Understand Easy 2 MARKS

Q7. A coin is flipped 200 times. Heads appears 108 times. (a) What is the experimental P(heads)? (b) The theoretical P(heads) is 0.5. Does this difference suggest the coin is biased? Explain.

Answer in your workbook.
Reason Hard 4 MARKS

Q8. Explain the law of large numbers in your own words. Include: (a) what it states, (b) one real-life example where it applies, and (c) why a doctor conducting a medical trial would use 1000 patients rather than 10.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B12 ÷ 40 = 3/10.

2. C It approaches (gets closer to) theoretical probability.

3. A Relative frequency.

4. D1/6 × 300 = 50.

5. B Random variation in a small sample.

6. B Student B's estimate is more reliable. With only 8 draws, Student A's 75% is easily explained by random variation, but Student B's 200 draws make their 71% much closer to the bag's true probability (law of large numbers).

Show Your Working Model Answers

Q6 (3 marks): P(red) = 22/60 [1] = 11/30 [1] ≈ 0.367 = 36.7% [1].

Q7 (2 marks): (a) P(heads) = 108/200 = 0.54 [1]. (b) Not necessarily biased, 0.54 vs 0.5 is a small difference with 200 trials. Random variation is expected. You would need many more trials to be confident [1].

Q8 (4 marks): (a) As the number of trials increases, the experimental probability gets closer to the theoretical probability [1]. (b) Any valid example e.g. casino knows that over thousands of games, house edge guarantees profit even if individual games vary [1]. (c) With 10 patients, results are very unreliable, random variation could make an ineffective treatment look effective or vice versa. With 1000 patients, results much more closely reflect the true effectiveness of the drug [2].

Stretch Challenge · +25 XP, +10 coins

The Biased Coin Investigation

You suspect a coin is biased toward heads. Design a fair experiment to test this. Include: (1) How many trials you would run and why, (2) What results would make you confident the coin IS biased, (3) What results would make you confident it is NOT biased, (4) Why you can never be 100% certain.

Reveal solution

(1) At least 500–1000 trials to reduce random variation, small samples can give misleading results. (2) Consistently getting 60%+ heads across 1000+ trials suggests bias. (3) Results staying close to 50% across many trials suggest it's fair. (4) Because any sequence of results is theoretically possible from a fair coin, probability never gives certainty, only likelihood.

R
Quick Review

Formula

Exp. P = frequency ÷ total trials

Relative frequency

Same thing as experimental probability

Law of large numbers

More trials → closer to theoretical

Small samples

High variation, don't trust them alone

Expected frequency

Theoretical P × number of trials

All RF add to 1

Relative frequencies for all outcomes sum to 1

Interactive: Coin Flip Simulator

See the law of large numbers in action. Flip a virtual coin and watch how the experimental probability settles toward 0.5 as you do more and more flips.

Practice alternative: Use the worked example and worksheet questions on this page for the same learning goal.

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