Mathematics · Simulation and virtual lab Free
Probability: from experiments to theory
Repeat an experiment 1, 10, 100 or 1 000 times and watch the relative frequency get closer to the probability.
- You will learn to
- Repeat an experiment many times and record relative frequencies.
- Compare relative frequency with theoretical probability.
- Explain the law of large numbers.
- You will measure
- Number of successes and relative frequency after 10 to 1 000 trials.
- Time needed
- About 20–30 min
- Equipment
- Nothing extra. Real version: coins, dice, or beans in a basket.
- Price
- Free: no payment, no account.
Virtual lab
Follow the steps with the simulation above (go back to the simulation ↑). Each time you tap Record, a row is added to the table and a point to the graph.
Aim
To compare the relative frequency of an event with its theoretical probability as the number of trials increases.
What to do
- Choose Toss a coin. Tap × 10 and record. Is the relative frequency of heads exactly 0.5?
- Tap × 100 several times and record after 100, 500 and 1 000 trials. What happens to the difference?
- Choose Roll a die and repeat. The event is “a six”, with P = 1/6.
- Choose Roll two dice: total. Which total is most common? Why?
- Choose the bean basket. Predict P(red), then test it with 1 000 draws.
- Write a conclusion about relative frequency and probability.
Results
Questions
What is the probability of a six with a fair die?
P(6) = 1/6 ≈ 0.167, because one of the six equally likely outcomes is a six.
Why is the total 7 the most likely with two dice?
7 can be made in 6 ways out of 36: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So P(7) = 6/36 = 1/6. Totals like 2 or 12 can be made in only one way.
After 10 tosses a learner got 7 heads. Is the coin unfair?
Not necessarily. With a small number of trials the relative frequency varies a lot. After 1 000 tosses it should be close to 0.5.
Find P(red), P(white) and P(not black) for the bean basket.
P(red) = 5/10 = 0.5, P(white) = 3/10 = 0.3, P(not black) = 1 − 2/10 = 0.8.
Two beans are drawn with replacement. Find P(both red).
P = 0.5 × 0.5 = 0.25, because with replacement the draws are independent.
What is the law of large numbers?
As the number of trials increases, the relative frequency of an event gets closer and closer to its theoretical probability.
For teachers: expected results and how the model works
Results are truly random (computer random numbers), so every class gets different numbers: compare tables between groups. For S2, ask learners to draw tree diagrams for two coins or two beans and test them. For S6, use the two-dice totals to discuss the expected value E(X) = 7.
Exam practice Exam set
National-exam style questions on this simulation (16 marks). Try each one on paper first, then open the marking scheme.
A fair die is thrown 600 times. (a) What is the probability of getting a 6? (b) How many sixes would you expect? (c) A learner got 130 sixes. Is the die likely to be fair? Explain. (5 marks)
Marking scheme: (a) 1/6 (1). (b) 600 × 1/6 = 100 (1). (c) Relative frequency = 130 ÷ 600 = 0.217 (1), noticeably more than 0.167 after many throws (1), so the die may be biased towards 6 (some variation is normal; more throws would confirm) (1).
Two coins are tossed. (a) List the sample space. (b) Find the probability of getting exactly one head. (3 marks)
Marking scheme: (a) HH, HT, TH, TT (1). (b) 2 out of 4 outcomes (1) = 1/2 (1).
Two dice are thrown and the scores added. Find the probability that the total is 7. Explain why 7 is the most likely total. (4 marks)
Marking scheme: 36 equally likely outcomes (1); six give 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) (1); P = 6/36 = 1/6 (1). No other total can be made in as many ways (1).
Explain the law of large numbers using the results of a coin-tossing experiment. (2 marks)
Marking scheme: As the number of trials increases (1), the relative frequency of heads gets closer and closer to the theoretical probability 0.5 (1).
A basket has 3 red and 5 white beans. One bean is taken at random, replaced, and a second is taken. Find the probability that both are red. (2 marks)
Marking scheme: P(red) = 3/8 each time (1); P(both red) = 3/8 × 3/8 = 9/64 (1).
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Probability: from experiments to theory
Aim: To compare the relative frequency of an event with its theoretical probability as the number of trials increases.
Apparatus (real laboratory): coins, dice, or beans in a basket.
What to do (with the simulation on www.donatstem.co.rw)
- Choose Toss a coin. Tap × 10 and record. Is the relative frequency of heads exactly 0.5?
- Tap × 100 several times and record after 100, 500 and 1 000 trials. What happens to the difference?
- Choose Roll a die and repeat. The event is “a six”, with P = 1/6.
- Choose Roll two dice: total. Which total is most common? Why?
- Choose the bean basket. Predict P(red), then test it with 1 000 draws.
- Write a conclusion about relative frequency and probability.
Results
| # | Experiment | Event | Trials | Frequency | Relative frequency | Probability |
|---|
| 1 | | | | | | |
| 2 | | | | | | |
| 3 | | | | | | |
| 4 | | | | | | |
| 5 | | | | | | |
| 6 | | | | | | |
| 7 | | | | | | |
| 8 | | | | | | |
Graph
Plot the relative frequency (vertical axis) against the number of trials. Draw a horizontal line at the theoretical probability.
Questions
- What is the probability of a six with a fair die? [2]
- Why is the total 7 the most likely with two dice? [2]
- After 10 tosses a learner got 7 heads. Is the coin unfair? [2]
- Find P(red), P(white) and P(not black) for the bean basket. [2]
- Two beans are drawn with replacement. Find P(both red). [2]
- What is the law of large numbers? [2]
- A fair die is thrown 600 times. (a) What is the probability of getting a 6? (b) How many sixes would you expect? (c) A learner got 130 sixes. Is the die likely to be fair? Explain. [5]
- Two coins are tossed. (a) List the sample space. (b) Find the probability of getting exactly one head. [3]
- Two dice are thrown and the scores added. Find the probability that the total is 7. Explain why 7 is the most likely total. [4]
- Explain the law of large numbers using the results of a coin-tossing experiment. [2]
- A basket has 3 red and 5 white beans. One bean is taken at random, replaced, and a second is taken. Find the probability that both are red. [2]
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Marking scheme: Probability: from experiments to theory
Expected results
Results are truly random (computer random numbers), so every class gets different numbers: compare tables between groups. For S2, ask learners to draw tree diagrams for two coins or two beans and test them. For S6, use the two-dice totals to discuss the expected value E(X) = 7.
Answers
- What is the probability of a six with a fair die? [2]
P(6) = 1/6 ≈ 0.167, because one of the six equally likely outcomes is a six.
- Why is the total 7 the most likely with two dice? [2]
7 can be made in 6 ways out of 36: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So P(7) = 6/36 = 1/6. Totals like 2 or 12 can be made in only one way.
- After 10 tosses a learner got 7 heads. Is the coin unfair? [2]
Not necessarily. With a small number of trials the relative frequency varies a lot. After 1 000 tosses it should be close to 0.5.
- Find P(red), P(white) and P(not black) for the bean basket. [2]
P(red) = 5/10 = 0.5, P(white) = 3/10 = 0.3, P(not black) = 1 − 2/10 = 0.8.
- Two beans are drawn with replacement. Find P(both red). [2]
P = 0.5 × 0.5 = 0.25, because with replacement the draws are independent.
- What is the law of large numbers? [2]
As the number of trials increases, the relative frequency of an event gets closer and closer to its theoretical probability.
- A fair die is thrown 600 times. (a) What is the probability of getting a 6? (b) How many sixes would you expect? (c) A learner got 130 sixes. Is the die likely to be fair? Explain. [5]
(a) 1/6 (1). (b) 600 × 1/6 = 100 (1). (c) Relative frequency = 130 ÷ 600 = 0.217 (1), noticeably more than 0.167 after many throws (1), so the die may be biased towards 6 (some variation is normal; more throws would confirm) (1).
- Two coins are tossed. (a) List the sample space. (b) Find the probability of getting exactly one head. [3]
(a) HH, HT, TH, TT (1). (b) 2 out of 4 outcomes (1) = 1/2 (1).
- Two dice are thrown and the scores added. Find the probability that the total is 7. Explain why 7 is the most likely total. [4]
36 equally likely outcomes (1); six give 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) (1); P = 6/36 = 1/6 (1). No other total can be made in as many ways (1).
- Explain the law of large numbers using the results of a coin-tossing experiment. [2]
As the number of trials increases (1), the relative frequency of heads gets closer and closer to the theoretical probability 0.5 (1).
- A basket has 3 red and 5 white beans. One bean is taken at random, replaced, and a second is taken. Find the probability that both are red. [2]
P(red) = 3/8 each time (1); P(both red) = 3/8 × 3/8 = 9/64 (1).