If I have rolled a 4 on a die 8 times in succession, the probability of rolling a 4 again is still only ≈ 0.16 (assuming the die is fair). Probability does not tell you what will happen next but predicts what will happen in the long run.
Step-by-step solution
Idea: Each roll of a fair die is independent: the die does not remember earlier rolls, so the chance of each face stays 16 on every roll. Probability is about what happens over many trials, not about one particular trial.
- Next roll: the die still has 6 equally likely faces and only one of them is a 4. So P(4 on the next roll) = 16 = 0.1666…. (Cutting off the decimals gives the ≈ 0.16 in the statement; rounded to two places it is 0.17. Both describe 16.)½ mark
- Why the past does not matter: a die is a cube of plastic; it cannot remember that it showed 4 eight times. Each roll is a fresh start, so earlier results do not make a 4 more likely or less likely. Thinking otherwise is called the Gambler’s Fallacy.½ mark
- What probability does tell us: over a long run of rolls, about 16 of them will be 4s. In 600 rolls, expect about 16 × 600 = 100 fours; in 6000 rolls, about 1000. It cannot tell us the result of the very next roll.½ mark
- (If a die really kept giving 4 hundreds of times in a row, we would start to doubt that it is fair. That is why the statement says “assuming the die is fair”.)½ mark
Check: A quick simulation idea: in a long list of die rolls, look only at the rolls that come right after a 4. About 16 of those are also 4s, the same as for all rolls.
Answer to write in the exam
Fair die: S = {1, 2, 3, 4, 5, 6}, equally likely; each roll is independent.
P(4 on the next roll) = 16 = 0.1666…
Past results do not change this (no memory).
In the long run: in 600 rolls, about 16 × 600 = 100 fours.
∴ The statement is true: P(4) stays 16; probability predicts the long run, not the next roll.
Common mistakes that cost marks
- Thinking a 4 is now “less likely” because it has come up too often (or “more likely” because it is on a lucky streak). For a fair die the chance stays 16.
- Reading probability 16 as “exactly one 4 in every 6 rolls”. It is a long-run share; in 6 rolls you may get no 4 or several.
- Rounding 0.1666… to 0.166 or 0.16 and calling that “rounded”. To two decimal places it rounds to 0.17.
How this can come in the exam
A fair coin has shown tails on the last 5 tosses. The probability that the next toss shows tails is
- 0
- 132
- 12
- 1
Show answer
(C) 12
Each toss is independent; P(tails) stays 12.
Assertion (A): If a fair die is rolled 600 times, a 6 is expected to come up about 100 times.
Reason (R): If a fair die is rolled 6 times, a 6 must come up exactly once.
- 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
(C) A is true but R is false.
A is true: 16 × 600 = 100 is the long-run expectation. R is false: in 6 rolls there may be no 6 or more than one.
Try one yourself
A fair die is rolled 300 times. About how many times would you expect to get a 4? Will it be exactly this number?
Show answer
About 16 × 300 = 50 times. Not necessarily exactly 50; the actual count varies, but it is usually close to 50.
More questions like this
- Let us say you are playing Snakes and Ladders, and you are rolling a fair 6-sided die to move. You have just rolled the die three times in a row, and each time you got a 6. Now, you think: ‘I have already rolled three 6s — there is no way I will get a 6 again on the next roll!’
- A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts the number of sweets of each colour:
10 red sweets | 8 green sweets | 7 yellow sweets | 5 blue sweets - A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are:
14 students: Science Club | 11 students: Arts Club |
9 students: Sports Club | 6 students: Debate Club
Assume there are 800 students in the whole school. - Toss a coin 20 times and record the result each time (heads or tails).
- Toss a paper cup into the air 100 times. After each toss record whether the cup lands on its bottom, upside down on its top or on its side (See the figure). Assign probabilities to the outcomes by using experimental probability.