Experimental Probability
Calculate experimental probability from trial results, compare it against the theoretical value, and use the law of large numbers to judge whether a difference is meaningful.
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Think First
You flip a coin 10 times and get 7 heads. Does that mean the coin is biased? What if you did it 1000 times and got 700 heads?
Experimental Probability
Probability estimated from real trial data, as the number of trials grows, the result gets closer to the theoretical value.
What You'll Master
- Calculate experimental probability from trial data using the relative frequency formula
- Distinguish between experimental and theoretical probability and know when each applies
- Explain the law of large numbers and why more trials give more reliable estimates
- Design a simple simulation to estimate probability experimentally
Words You Need
⚠ Spot the Trap
Getting 7 heads from 10 flips does NOT prove the coin is biased, small samples are naturally noisy. Only large samples (hundreds or thousands of trials) can reliably suggest bias. Always compare the number of trials before drawing conclusions.
1. Experimental Probability Formula
Experimental probability (relative frequency) is calculated directly from data:
$$P(\text{event}) \approx \frac{\text{number of times event occurred}}{\text{total number of trials}}$$Example: A coin is flipped 40 times. Heads appears 22 times.
$$P(\text{H}) \approx \frac{22}{40} = \frac{11}{20} = 0.55$$The theoretical probability is $0.5$. The experimental result is close but not exactly equal, this is normal variation for a small sample.
2. Experimental vs Theoretical Probability
Both express the likelihood of an event but they come from different sources:
- Theoretical: calculated by counting equally likely outcomes. Requires no experiment. Example: $P(\text{rolling a 4}) = \frac{1}{6} \approx 0.167$.
- Experimental: measured by actually running trials. Results vary. Example: rolling a die 60 times and recording how often a 4 appears.
We use theoretical probability when outcomes are truly equally likely. We use experimental probability when we do not know the theoretical value (e.g., a biased die, weather forecasting, medical testing).
3. The Law of Large Numbers
The more trials we run, the closer the experimental probability gets to the theoretical probability. This is called the law of large numbers.
- 10 flips of a fair coin: might give 3 heads ($P \approx 0.30$), very noisy
- 100 flips: likely to give between 40–60 heads ($P \approx 0.40\text{–}0.60$)
- 10,000 flips: very likely to give close to 5,000 heads ($P \approx 0.50$)
This is why large studies are more reliable than small ones, and why casinos always profit over many rounds even if a player wins in the short term.
4. Simulations
A simulation models a real situation using a random device. Choose your simulation tool so that the outcomes match the probabilities you want.
Example: Simulate rolling a die 60 times to estimate $P(\text{prime})$.
Prime outcomes on a die: $\{2, 3, 5\}$, 3 out of 6, so theoretical $P(\text{prime}) = \frac{3}{6} = 0.5$.
Record results in a frequency table:
| Outcome | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Frequency | 9 | 11 | 10 | 8 | 12 | 10 |
Prime frequency: $11 + 10 + 12 = 33$. $P(\text{prime}) \approx \dfrac{33}{60} = 0.55$, close to theoretical $0.5$. ✓
Common Pitfalls
- Concluding bias from too few trials, small samples are always noisy
- Using the wrong denominator, it must be the total number of trials, not the number of outcomes
- Expecting experimental probability to exactly equal theoretical, they will differ, especially for small samples
- Designing a simulation with unequal probabilities when equal probabilities are needed (e.g., using 1–3 for "yes" and 4–6 for "no" is fine; 1–3 for "yes" and 4–5 for "no" is biased)
Copy This Into Your Book
$$P_{\text{experimental}}(\text{event}) \approx \frac{\text{frequency of event}}{\text{total trials}}$$Law of large numbers: as trials increase, experimental probability $\to$ theoretical probability.
Simulation: use a random device whose outcomes match the required probabilities to model a real experiment.
A die is rolled 60 times and the outcome 3 appears 15 times. What is the experimental probability of rolling a 3?
As the number of trials in an experiment increases, the experimental probability:
Which type of probability is calculated by counting equally likely outcomes without running any experiment?
The law of large numbers states that:
A spinner has a $\frac{1}{4}$ chance of landing on red. In 200 spins, how many times would you expect red to appear?
Q6. A die is rolled 120 times. The number 5 appears 18 times. (a) Calculate the experimental probability of rolling a 5. (b) Compare to the theoretical probability. (c) Is there evidence the die is biased? Explain your reasoning.
Q7. Design a simulation using a die to model a multiple-choice quiz where each question has 4 options (A, B, C, D) with equal probability. (a) Describe which die outcomes represent each option. (b) Explain how you would use this to simulate 20 questions and find the probability of guessing correctly.
Q8. A spinner has 4 equal sectors labelled 1–4. It is spun 200 times with these results: 1 appears 46 times, 2 appears 54 times, 3 appears 48 times, 4 appears 52 times. (a) Calculate the experimental probability for each outcome. (b) Compare to theoretical. (c) Does the spinner appear fair?
Show Answers
Q6
(a) $P(\text{rolling 5}) \approx \dfrac{18}{120} = \dfrac{3}{20} = 0.15$
(b) Theoretical: $P(\text{rolling 5}) = \dfrac{1}{6} \approx 0.167$
(c) The result (0.15) is reasonably close to theoretical (0.167). With only 120 trials this difference is within normal variation, no strong evidence of bias.
Q7
Use a 6-sided die. Assign: 1→A, 2→B, 3→C, 4→D, and re-roll when 5 or 6 appears (since we need 4 equal outcomes). Roll the die 20 times (re-rolling 5s and 6s). Each roll represents one question. $P(\text{correct}) = \frac{1}{4}$ theoretically. Count how many times the designated "correct" face appears.
Q8
(a) $P(1) \approx \frac{46}{200} = 0.230$; $P(2) \approx \frac{54}{200} = 0.270$; $P(3) \approx \frac{48}{200} = 0.240$; $P(4) \approx \frac{52}{200} = 0.260$
(b) Theoretical for each: $\frac{1}{4} = 0.25$
(c) All values are close to 0.25, the spinner appears fair. Variation is expected with 200 trials.
Stretch Challenge
Two students each roll a die 30 times. Student A gets $P(6) = \frac{8}{30} \approx 0.267$. Student B gets $P(6) = \frac{3}{30} = 0.10$. Neither is close to the theoretical $\frac{1}{6} \approx 0.167$.
(a) Explain why this is expected with only 30 trials each.
(b) If both students combine their results, calculate the combined experimental $P(6)$.
(c) Is the combined result closer to $\frac{1}{6}$? Why?