Suppose you roll a 6-sided die 12 times and get a ‘3’ three times.
- (i) What is the experimental probability of rolling a ‘3’?
- (ii) What is the theoretical probability of rolling a ‘3’?
- (iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?
Step-by-step solution
Idea: Experimental probability comes from the results; theoretical probability from equally likely outcomes. They can differ, especially for few trials; the Law of Large Numbers says they get closer as the number of trials grows.
(i) What is the experimental probability of rolling a ‘3’?
- Experimental P(3) = Number of times 3 occurredTotal number of rolls = 312½ mark
- = 14 = 0.25.½ mark
(ii) What is the theoretical probability of rolling a ‘3’?
- For a fair die the outcomes {1, 2, 3, 4, 5, 6} are equally likely; one of them is 3.½ mark
- P(3) = 16 ≈ 0.167.½ mark
(iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?
- Why different: experimental probability depends on what actually happened, and each roll is random. With only 12 rolls, getting a 3 just once more or less than expected changes the result a lot (theory expects 16 × 12 = 2 threes; we got 3).1 mark
- More rolls: as the number of rolls increases, the experimental probability tends to get closer to the theoretical 16. This is the Law of Large Numbers. We would expect about 16 × 60 = 10 threes in 60 rolls, about 100 in 600 and about 1000 in 6000, and the relative frequency would be closest to 16 ≈ 0.167 for 6000 rolls.1 mark
Check: 0.25 − 0.167 ≈ 0.083, which is just one extra 3 in 12 rolls (112 ≈ 0.083). A small difference in count makes a big difference in probability when the number of trials is small.
Answer to write in the exam
(i)
Experimental P(3) = Number of times 3 occurredTotal rolls = 312
∴ Experimental P(3) = 14 = 0.25
(ii)
S = {1, 2, 3, 4, 5, 6}; favourable = 1
∴ Theoretical P(3) = 16 ≈ 0.167
(iii)
12 is a small number of trials, so the experimental result varies a lot by chance.
Expected number of 3s: 16 × 60 = 10; 16 × 600 = 100; 16 × 6000 = 1000.
∴ As the number of rolls increases, the experimental probability gets closer to 16 (Law of Large Numbers).
Common mistakes that cost marks
- Concluding from 12 rolls that the die is unfair. 3 threes in 12 rolls happens quite often with a fair die.
- Writing that with 6000 rolls you will get exactly 1000 threes. You expect about 1000; the relative frequency will be close to 16, not necessarily equal.
- Mixing up the two answers: the experimental probability uses the 12 rolls (312); the theoretical one does not (16).
How this can come in the exam
Assertion (A): A die rolled 12 times gave an experimental probability of 14 for getting a 3, although the theoretical probability is 16.
Reason (R): Experimental probability can differ from theoretical probability, especially when the number of trials is small.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Both are true, and R explains A: 12 rolls is a small number of trials, so the experimental value (312 = 14) can be quite far from 16.
A die is rolled 30 times and a 6 comes up 8 times. Find the experimental and theoretical probabilities of getting a 6, and state which would change if the die were rolled 3000 times.
Show answer
Experimental: 830 = 415 ≈ 0.27. Theoretical: 16 ≈ 0.17. The theoretical value stays 16; the experimental value would change and would be expected to come much closer to 16 after 3000 rolls.Try one yourself
A coin is tossed 10 times and shows heads 7 times. Find the experimental and theoretical probabilities of heads. What would you expect after 1000 tosses?
Show answer
Experimental 710 = 0.7; theoretical 12 = 0.5. After 1000 tosses the experimental probability should be close to 0.5 (about 500 heads).
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