Practice Maths

L19 — Investigating Probabilities

Theoretical vs Experimental Probability

In this lesson, we look at how actual data (experimental probability) compares to our mathematical predictions (theoretical probability), and how the size of an experiment affects its reliability.

Key Terms

Law of Large Numbers
as the number of trials increases, experimental probability gets closer and closer to theoretical probability
relative frequency
the number of times an event occurs divided by the total number of trials; used as experimental probability
biased
describes an experiment or object where outcomes are not equally likely due to a physical or design flaw (e.g. a weighted die)
prediction
an estimate of what is likely to happen in the future, based on theoretical or experimental probability
The Law of Large Numbers
The larger the number of trials, the closer experimental probability gets to theoretical probability. More data = more reliable results.

Worked Example

Question: A coin is flipped 20 times and lands Heads 13 times.

Step 1 — Calculate experimental probability: P(Heads) = 1320 = 0.65

Step 2 — State theoretical probability: P(Heads) = 12 = 0.5

Step 3 — Compare: Experimental (0.65) differs from theoretical (0.5). This is normal with only 20 trials — more trials would bring them closer together.

Theoretical vs Experimental Probability

Theoretical probability is what we calculate using maths: P(event) = favourable outcomes ÷ total outcomes. It’s what should happen in a perfectly fair situation.

Experimental probability (also called relative frequency) is what actually happens when you run a real experiment. You record the results and calculate: frequency of event ÷ total trials.

For example, a fair coin should theoretically give 50% heads. But if you flip it 10 times, you might get 6 heads (60%). The experimental probability doesn’t always match the theoretical — and that’s normal!

Why More Trials Gives Better Results

The more times you repeat an experiment, the closer your experimental probability gets to the theoretical probability. This is called the Law of Large Numbers.

  • 10 coin flips: you might get 7 heads (70%) — quite different from 50%
  • 100 coin flips: you might get 52 heads (52%) — getting closer
  • 1000 coin flips: you might get 503 heads (50.3%) — very close to 50%

So when investigating probability, more data always gives more reliable results.

Recording and Displaying Results

When running a probability experiment, use a tally table to record your results. After the experiment, calculate the relative frequency for each outcome:

  • Roll a die 30 times. Record how many times each number (1–6) appears.
  • Relative frequency of rolling a 4 = (number of 4s) ÷ 30
  • Theoretical probability of a 4 = 16 ≈ 0.167
  • Compare: are they close? The larger your sample, the closer they should be.

Designing a Probability Experiment

A good probability experiment needs:

  • A clear question: “Is this coin fair?” or “Which colour marble is most likely?”
  • A method: Describe exactly what you will do (flip the coin, record the result, repeat).
  • Enough trials: At least 20–30 for a classroom experiment; more is always better.
  • A comparison: Compare your experimental results to the theoretical probability.
Key tip: If your experimental probability is very different from the theoretical probability after many trials, this might suggest the experiment isn’t fair (e.g. a biased coin or weighted die). But with very few trials, a large difference is perfectly normal — never draw conclusions from small samples alone.

Practice Questions

  1. Definitions Check Fluency

    Complete the following sentences:

    1. The probability calculated using maths (logic) is called __________ probability.
    2. The probability calculated from actual results is called __________ probability.
    3. Another name for experimental probability is __________ __________.
    4. The __________ __ __________ __________ states that as the number of trials increases, experimental probability gets closer to theoretical probability.
    5. One single flip of a coin in an experiment is called a __________.

  2. The 5-Section Spinner Fluency

    A fair spinner has 5 equal sections: Red, Blue, Green, Yellow, and Orange.

    R B G Y O Red Blue Green Yellow Orange
    1. What is the theoretical probability of landing on Blue? Write as a fraction, decimal, and percentage.
    2. If you spin it 100 times, how many times would you theoretically expect each colour to appear?
    3. You spin it 50 times and land on Green 12 times. Calculate the experimental probability of Green as a decimal.
    4. Is the experimental probability of Green from (c) higher or lower than the theoretical probability?
    5. With 500 spins instead of 50, would you expect the experimental probability to be closer to or further from the theoretical probability? Explain why.

  3. Analysing Trial Size Fluency

    A computer simulates flipping a coin. Here are the results for Heads:

    TrialsHeads CountRelative Frequency
    10770%
    1005656%
    100050450.4%
    1. What is the theoretical probability of Heads as a percentage?
    2. Which trial size gives the result closest to the theoretical probability?
    3. Why does a larger number of trials produce a more reliable result?
    4. If 10,000 trials were used, would you expect the result to be closer to or further from 50%? Explain.
    5. Is it possible for relative frequency to ever be exactly equal to 50% in a real experiment? Explain.

  4. Predicting Future Events Fluency

    A soccer player has scored 12 goals out of 30 penalty kicks in her career.

    1. Write her relative frequency as a fraction in simplest form.
    2. Write this as a decimal and a percentage.
    3. If she takes 50 more penalty kicks, how many goals would you predict she will score, based on her data?
    4. After a further 20 kicks she scores 6 more goals (total: 18 goals out of 50 kicks). Calculate her new relative frequency as a fraction in simplest form.
    5. Did her relative frequency go up or down compared to her original result?

  5. The Points Race Understanding

    A game is played with a standard 6-sided die.
    Player A scores 5 points whenever an odd number is rolled.
    Player B scores 10 points whenever a 6 is rolled.

    1. What is the theoretical probability of Player A scoring on any roll? Write as a fraction.
    2. What is the theoretical probability of Player B scoring on any roll? Write as a fraction.
    3. Who is more likely to score on any given roll?
    4. Player B earns double the points. Is the game fair? Use probability to justify your answer.

  6. Investigating the Marble Bag Understanding

    A bag contains 3 black marbles and 7 white marbles.

    1. Calculate the theoretical probability of picking a black marble. Write as a decimal.
    2. In 20 trials (replacing the marble each time), a student picked black 4 times. Calculate the experimental probability of black.
    3. Is the experimental probability above or below the theoretical probability?
    4. If the experiment were repeated with 200 trials, would you expect the experimental probability to be closer to or further from the theoretical probability? Explain.

  7. Equally Likely or Not? Understanding

    1. Is rolling a standard 6-sided die an equally likely experiment? Explain.
    2. Is predicting whether the next car that drives past will be red or any other colour an equally likely experiment? Explain.
    3. Give one example of a probability experiment that is equally likely and explain why.
    4. A spinner has four sections of unequal size. What can you say about the probability of landing on the largest section compared to the smallest?

  8. Relative Frequency in Context Understanding

    A study of 200 bus journeys found that 40 buses were late.

    1. Calculate the relative frequency of a bus being late. Write as a fraction in simplest form.
    2. Write this as a percentage.
    3. If 150 buses run on a Monday, how many would you predict will be late, based on this data?
    4. Why would relative frequency based on 2000 bus journeys be more reliable for prediction than 200? Explain.

  9. The Shape Bag Investigation Problem Solving

    A container holds 8 triangles, 4 squares, 5 circles, and 3 hexagons (20 shapes total). Shapes are drawn at random and replaced each time.

    1. Calculate the theoretical probability of drawing a square OR a circle. Write as a decimal.
    2. In 50 trials, the student drew a square or circle 26 times. Calculate the experimental probability.
    3. Is the experimental probability higher or lower than theoretical? Is this evidence the experiment was unfair? Explain.

  10. Investigating a Biased Die Problem Solving

    Jack thinks his 6-sided die lands on 6 more often than expected. He rolls it 300 times and records 85 sixes.

    1. How many sixes would Jack theoretically expect in 300 rolls? Show working.
    2. Calculate the experimental probability of rolling a 6 from Jack’s results. Write as a percentage.
    3. Is there evidence that the die is biased? Compare the experimental and theoretical probabilities and justify your conclusion.
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