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!’
Step-by-step solution
To find: Whether a 6 on the next roll is really impossible, and its probability
Idea: A die has no memory. Each roll of a fair die is an independent event, so earlier results do not change the probability of any face.
- The die is fair, so on any roll the 6 faces are equally likely. Favourable outcomes for a 6: 1. Possible outcomes: 6.½ mark
- Probability of rolling a 6 = 16 = 0.1666… ≈ 0.167, about 16.7%. This is the same on every roll.½ mark
- Each roll of the die is an independent event: it does not change based on what happened in previous turns. So three 6s in a row do not make the next 6 impossible, or even less likely.½ mark
- Believing that a run of one result makes it less likely next time is the Gambler’s Fallacy. In Snakes and Ladders, as in many games of chance, randomness has no memory.½ mark
- You may also see this written as ≈ 0.166 (16.6%), with the later digits simply cut off. Rounded to three decimal places, 16 = 0.1666… ≈ 0.167.
Check: If three 6s really made a fourth impossible, the die would need to “know” its past rolls. A fair die is just a symmetrical cube, so it cannot; the chance stays 16.
Answer to write in the exam
Fair die; each roll is an independent event.
P(6) = 16 ≈ 0.167 on every roll
∴ The thinking is wrong (Gambler’s Fallacy); P(6 on the next roll) = 16 ≈ 16.7%
Common mistakes that cost marks
- Saying the probability of a 6 is now 0 because “6 has had its turn”. Past rolls do not change it; it is still 16.
- Multiplying or dividing 16 by the number of earlier 6s (e.g. 118). The next roll has its own probability 16.
- Thinking the opposite mistake, that a 6 is “on a roll” and more likely next time. For a fair die it is neither more nor less likely.
How this can come in the exam
Assertion (A): A fair die showed 2 on each of its last four rolls, so the probability of a 2 on the next roll is less than 16.
Reason (R): Each roll of a fair die is independent of earlier rolls.
- 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
(D) A is false but R is true.
A is false (the probability is still 16); R is true, and it is why A is false.
Meena has lost the toss in her last six matches. She says, “I am sure to win the toss today.” Is she right? Give the probability she wins today’s toss.
Show answer
No. Each toss of a fair coin is independent of earlier tosses; the coin has no memory. Her chance of winning today’s toss is still 12. Believing a win is “due” is the Gambler’s Fallacy.Try one yourself
A fair spinner has 4 equal sectors numbered 1 to 4. It has landed on 3 five times in a row. What is the probability it lands on 3 next time?
Show answer
14 = 0.25. The spins are independent, so past results do not change it.
More questions like this
- 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.
- What is the probability of getting an even number when rolling a fair 6-sided die?