Practice Maths

Experimental vs Theoretical Probability

Key Ideas

Key Terms

Theoretical probability
Calculated using symmetry or equally likely outcomes without performing the experiment: P(event) = (number of favourable outcomes) / (total outcomes).
Experimental probability
(relative frequency) is calculated from data collected by actually performing the experiment: P(event) ≈ (number of times event occurred) / (number of trials).
law of large numbers
As the number of trials increases, experimental probability gets closer to theoretical probability.
fair
If all outcomes are equally likely. It is biased if some outcomes are more likely than others.
Hot Tip Even a fair coin might show 7 heads in 10 flips — this does not prove bias. With only 10 trials, variation is expected. Always consider the number of trials before concluding something is unfair.

Worked Example

Question: A six-sided die is rolled 120 times. The number 4 appears 28 times. (a) What is the theoretical probability of rolling a 4? (b) What is the experimental probability from these trials? (c) Does this suggest the die is biased?

(a) Theoretical: P(4) = 1/6 ≈ 0.1667

(b) Experimental: P(4) = 28/120 = 7/30 ≈ 0.233

(c) Expected frequency: 1/6 × 120 = 20. Observed: 28 — this is 40% higher than expected. With 120 trials, this discrepancy is notable and may suggest the die is slightly biased towards 4, though more trials would be needed to be certain.

Theoretical Probability

Theoretical probability is calculated using logic and the assumption that all outcomes are equally likely. It does not require you to perform any experiment.

Formula: P(event) = (number of favourable outcomes) ÷ (total number of equally likely outcomes)

Example: Rolling a standard die, the theoretical probability of getting a 4 is 1/6 because there is 1 favourable outcome out of 6 equally likely outcomes. The theoretical probability of rolling an even number is 3/6 = 1/2.

Theoretical probability is exact and reproducible — it gives the same answer every time because it is based on logic, not chance.

Experimental (Relative Frequency) Probability

Experimental probability (also called relative frequency) is based on what actually happened when an experiment was conducted.

Formula: Experimental P(event) = (number of times event occurred) ÷ (total number of trials)

Example: A coin is flipped 80 times and lands heads 43 times. The experimental probability of heads = 43/80 = 0.5375. This is close to, but not exactly equal to, the theoretical probability of 1/2.

Experimental probability varies every time you repeat the experiment because it depends on actual random outcomes.

The Law of Large Numbers

The Law of Large Numbers states that as the number of trials increases, the experimental probability gets closer and closer to the theoretical probability.

With only 10 coin flips you might get 7 heads (experimental prob = 7/10 = 0.7), which is well above the theoretical 0.5. But with 10 000 flips, you would expect very close to 5000 heads. The larger the number of trials, the more reliable the experimental probability becomes.

This is why large sample sizes are valued in science and medicine — more trials mean more reliable results.

Simulations

A simulation models a real-world probability situation using a simpler random device — a coin, a die, random number tables, or a computer. Simulations are useful when the real experiment is expensive, dangerous, time-consuming, or impossible to repeat many times.

Example: To simulate the probability that a baby is born with a particular genetic trait (probability 1/4), you could roll a four-sided die and let "1" represent the trait. Roll it 200 times and count how often you roll a 1 — this estimates the probability. Alternatively, use a random number generator: let digits 0–2 represent the trait (probability 3/10 ≈ correct if trait probability is 0.3).

When designing a simulation, clearly state: (1) what the random device is, (2) which outcomes represent each real-world event, and (3) how many trials you will run.

Comparing Theoretical and Experimental Results

If experimental results differ significantly from theoretical ones, this might indicate the experiment was biased (e.g. an unfair coin or die) — or it might just be due to a small sample size. More trials are needed to distinguish between bias and natural variation.

Key tip: Remember that "experimental probability equals theoretical probability" is not guaranteed — it is what we expect to approach with many trials. Don't be surprised if a small experiment gives results that differ from theory; comment that a larger number of trials would bring the results closer to the theoretical value.

Mastery Practice

  1. A standard six-sided die is rolled. Find the theoretical probability of each event. Fluency

    1. Rolling an even number.
    2. Rolling a number greater than 4.
    3. Rolling a prime number (2, 3, or 5).
    4. Rolling a 7.
    5. Rolling a number less than 7.

  2. A drawing pin is dropped 200 times. It lands point-up 74 times and point-down 126 times. Fluency

    1. Find the experimental probability of landing point-up.
    2. Find the experimental probability of landing point-down.
    3. Can you calculate a theoretical probability for a drawing pin? Explain.
    4. Based on these results, estimate how many times the pin would land point-up if dropped 500 times.

  3. A bag contains 4 red, 3 blue, and 3 green counters. A counter is drawn, noted, and returned. This is done 50 times. Fluency

    1. Find the theoretical probability of drawing red.
    2. Find the expected number of red counters drawn in 50 trials.
    3. Find the expected number of blue counters drawn.
    4. In the actual experiment, red appeared 23 times. Find the experimental probability of red.
    5. Is 23 a reasonable result? Explain using the expected frequency.

  4. A biased spinner with sectors A, B, C, D was spun 400 times. The results are shown below. Fluency

    OutcomeABCD
    Frequency841329688
    1. Verify the total frequency is 400.
    2. Find the experimental probability for each outcome (as a fraction and decimal).
    3. If the spinner were fair, what would be the theoretical probability for each outcome?
    4. Compare theoretical and experimental probabilities. Which sector appears to be favoured?

  5. A fair coin is flipped repeatedly. The cumulative proportion of Heads after each set of trials is recorded. Understanding

    Trials10501005001000
    Heads32247243491
    1. Calculate the experimental probability of Heads at each stage (as a decimal to 3 d.p.).
    2. What is the theoretical probability of Heads for a fair coin?
    3. Describe the trend in experimental probability as the number of trials increases.
    4. After 10 trials the proportion is only 0.3. Does this prove the coin is biased? Explain.
    5. State the law of large numbers in your own words.

  6. Anya suspects her spinner (sectors 1 to 4) is biased. She spins it 80 times. Understanding

    Sector1234
    Frequency16181432
    1. Find the experimental probability for each sector.
    2. If the spinner is fair, what is the expected frequency for each sector in 80 spins?
    3. Which sector shows the greatest difference from its expected frequency?
    4. Based on these results, does the spinner appear to be biased? Justify your answer.
    5. What would Anya need to do to be more confident in her conclusion?

  7. A jar contains red and blue marbles in an unknown ratio. A marble is drawn and replaced 60 times; red appeared 42 times. Understanding

    1. Estimate the probability of drawing red based on the experiment.
    2. Estimate the ratio of red to blue marbles in the jar.
    3. If you were told the jar contains 20 marbles, estimate how many are red.
    4. Could the theoretical probability of drawing red be exactly 2/3? Explain how you could check.
    5. How many more trials would you recommend to obtain a more reliable estimate? Explain.

  8. A class simulates rolling a die 300 times using a random number generator. Their results are shown. Understanding

    Face123456
    Frequency485346524952
    1. What was the expected frequency for each face?
    2. Find the experimental probability of rolling an even number from this simulation.
    3. Find the theoretical probability of rolling an even number.
    4. Comment on how well the simulation matches theory.

  9. A school canteen sells two brands of sandwich. Over 5 weeks (25 school days), the canteen records how many of each brand it sells each day. Brand A: average 48 sold per day. Brand B: average 32 sold per day. Total sandwiches sold per day: 80. Problem Solving

    1. Estimate the probability that a randomly selected sandwich sold is Brand A.
    2. If 200 sandwiches are stocked each day, how many of each brand should the canteen order to match expected demand?
    3. The canteen manager says “I should stock equal numbers of each brand.” Using the experimental probabilities, explain why this is not optimal.
    4. On a particular day, 60 students each independently buy a sandwich. Using the experimental probability, find the expected number of students who buy Brand A.

  10. A news article claims: “We tossed a coin 20 times and got 13 Heads. This proves the coin is biased!” Problem Solving

    1. Calculate the experimental probability of Heads from their trial.
    2. What is the theoretical probability for a fair coin?
    3. How many Heads would you expect in 20 flips of a fair coin?
    4. Explain why 13 Heads in 20 flips does NOT prove the coin is biased. What would be a stronger way to investigate?
    5. If the coin were flipped 2000 times and 1300 Heads were observed, would this be more convincing evidence of bias? Explain.
See Answers ➔