What does it mean?
Experimental probability summarises what happened: successes divided by trials. Theoretical probability describes a model. Eighteen heads in forty flips gives 18/40, while a fair-coin model assigns 1/2 to heads. Both statements can be reported without pretending they are the same kind of evidence.
Experimental probability, or relative frequency, is (number of successes) / (number of trials). Theoretical probability comes from an equally likely sample space. They are allowed to differ. In 40 fair-coin flips you might see 18 heads; 18/40 = 9/20, which is near 1/2 but not equal to it. That gap is variation. With more trials, the experimental value often moves closer to the theoretical value, but it need not match exactly. Do not change the sample-space model after one short run. Complements and two-way tables are a related next step.
Learning goal: Compare experimental and theoretical probability; explain variation. The grade is a suggested starting point; school sequences vary. Use the first worked example to check readiness, and Fractions if an earlier idea needs revisiting.
See the idea
Read each diagram with its labels, then follow the worked examples. Practice questions include their own measurements and information.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Choose a method and explain why it works
Define the event and trial, count occurrences and divide by the total trials. Simplify the fraction or convert to a decimal to compare with the model. For more data from the same experiment, combine success counts and trial counts before dividing; do not average percentages from unequal-sized runs.
Notation: P_exp = successes / trials. Compare with P_theory from the sample space.
Key rule and when to use it
Experimental probability = observed event count ÷ number of trials. A theoretical probability comes from the stated model, rather than the latest short run.
Worked examples, step by step
Example 1: A fair coin is flipped 40 times. Heads appears 18 times. Find the experimental probability of heads, and compare it with 1/2.
- Experimental probability = 18/40 = 9/20.
- Theoretical probability for a fair coin is 1/2 = 10/20.
- 9/20 is close to 10/20. The gap is variation in a short run, not proof that the coin is unfair.
Answer: 9/20; close to 1/2, not proof of bias
Check: 18 is 2 away from 20, which is half of 40, so the result is near the model.
Example 2: A spinner is equally likely to land on 1, 2, 3 or 4. In 200 spins, the result 1 appears 92 times. Find the experimental probability of 1 and the theoretical probability, then comment.
- This is a four-outcome spinner, not a coin. Theoretical P(1) = 1/4.
- Experimental P(1) = 92/200 = 23/50 = 0.46.
- The observed 0.46 differs from the model’s 0.25. This is a reason to investigate the setup or fairness assumption, not to silently replace the stated theoretical model.
Answer: 92/200; 1/4; the experimental value is not close to 1/4
Check: Half of 200 is 100, so 92 would be near 1/2, not near 1/4. That is why the comment is not "close enough".
Example 3: One run has 8 successes in 20 trials and a second has 32 in 80. Find the combined relative frequency.
- Combine successes: 8 + 32 = 40. Combine trials: 20 + 80 = 100.
- Relative frequency = 40/100 = 0.4.
- Combining counts uses each trial equally. For runs with different success rates, an unweighted average of their percentages can be misleading.
Common mistakes and important boundaries
More trials do not guarantee that every next running estimate gets closer to the theoretical value. Variation remains, though longer fair independent runs tend to be more stable. A short streak does not force the opposite result next, and zero observed successes does not by itself prove impossibility.
A point to explain
A learner may treat 18 heads in 40 flips as proof that the coin is not fair, or replace the theoretical value with the experimental one forever.
Compare an uncertain attempt with the closest worked example. Look at the meaning of each operation, unit and label before repeating the arithmetic.
Experimental Probability and Relative Frequency — practice
Work on paper first. Open each answer afterwards and compare the reasoning. Print the questions with or without the separate answer key; no account is required.
Name: ____________________ Date: __________
What is experimental probability?
Answer and a way to check
Successes divided by trials.
Count how many times the event happened. Divide by how many trials were run.
40 trials, 18 heads, fair coin. Which is the experimental probability of heads?
Answer and a way to check
18/40
Experimental uses the recorded 18 and 40, not the fair-coin model.
A die is rolled 60 times. A six appears 8 times. Find the experimental probability of a six as a fraction in lowest terms.
Answer and a way to check
2/15
Write 8/60, then divide numerator and denominator by 4.
For a fair die, what is the theoretical P(six) in lowest terms?
Answer and a way to check
1/6
A fair die has six faces. One of them is six.
80 fair-coin flips give 44 heads. Write the experimental P(heads) as a fraction in lowest terms.
Answer and a way to check
11/20
44/80. Divide numerator and denominator by 4.
A fair coin shows 3 heads in 10 flips. Why does that not prove the theoretical probability is 3/10?
Answer and a way to check
Ten trials is a short run. Variation can move the relative frequency away from 1/2 without changing the sample-space model.
Name the theoretical value from two equally likely faces. Then say that 10 is a small number of trials.
Fair spinner, four equal sectors. After 8 spins with no sector A, a student says P(A) is 0. What is the mistake?
Answer and a way to check
Eight trials can miss A by variation; theoretical P(A) is still 1/4.
Theoretical P(A) comes from four equal sectors. Eight spins can miss A.
A fair coin is flipped 200 times and lands heads 92 times. Write experimental P(heads) as a fraction in lowest terms, write theoretical P(heads), and say whether 92/200 is closer to 1/2 than 3 heads in 10 flips would be.
Answer and a way to check
23/50; 1/2; yes
92/200 simplifies by 4. Theoretical fair-coin value is 1/2. Compare the gaps 0.04 versus 0.2.
Experimental Probability and Relative Frequency — answer key
Successes divided by trials. Count how many times the event happened. Divide by how many trials were run.
18/40 Experimental uses the recorded 18 and 40, not the fair-coin model.
2/15 Write 8/60, then divide numerator and denominator by 4.
1/6 A fair die has six faces. One of them is six.
11/20 44/80. Divide numerator and denominator by 4.
Ten trials is a short run. Variation can move the relative frequency away from 1/2 without changing the sample-space model. Name the theoretical value from two equally likely faces. Then say that 10 is a small number of trials.
Eight trials can miss A by variation; theoretical P(A) is still 1/4. Theoretical P(A) comes from four equal sectors. Eight spins can miss A.
23/50; 1/2; yes 92/200 simplifies by 4. Theoretical fair-coin value is 1/2. Compare the gaps 0.04 versus 0.2.
Questions students and parents ask
Must the estimate get closer after every extra trial?
No. Relative frequency can move closer or farther after an individual trial. Larger independent samples tend to be more stable; exact agreement or steady improvement is not guaranteed.
Does five heads in a row make tails more likely next?
Not in the fair independent-coin model. The next toss still has probability 1/2 for tails; previous results do not create a debt to be repaid.
Is this a complete set of textbook exercise solutions?
No. This guide adapts BananaLearn’s own teaching examples and practice. It is not official NCERT material or a promise of the exact chapter sequence in a particular board, country or textbook edition.
What if the examples are still difficult?
Revisit Fractions. Try the first example with its model, then attempt one fresh question independently. A correct copied answer is not evidence that the method can be used without support.
Help at home
Record several runs before comparing them. Ask the learner to distinguish the observed fraction from the probability assumed by the model.
Keep one example of the learner’s own working. A repeated difficulty across fresh questions can guide a discussion with their teacher; one short practice sheet does not establish a diagnosis or a full mastery profile.
Connect this idea to the next skill
About this guide
Based on BananaLearn’s Grade 8 course lesson “Trials Versus the Model”. The explanations have been adapted for independent reading; course diagrams are reused where they fit the idea, and missing classroom givens are made explicit in questions.
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