A coin has no memory
Independent trials and the Gambler's Fallacy
Is tails due?
A fair coin shows heads six times in a row. It feels as if tails is now due. But does the coin remember what happened before?
After six heads in a row, the next toss is…
What this lesson covers
The idea
Each trial is independent: a coin or die has no memory, so past results never change the probability of the next outcome; believing otherwise is the Gambler's Fallacy. Probability predicts the long run, not the next result.
Is tails due?
A fair coin shows heads six times in a row. It feels as if tails is now due. But does the coin remember what happened before?
After six heads in a row, the next toss is…
- More likely tails
- Still equally likely heads or tails
- More likely heads
What follows a run?
The toy tosses a coin 20 000 times and finds every run of heads. It then counts what came next. Look at runs of 3 different lengths.
No memory
Each trial is independent: past results never change the probability of the next outcome. Thinking that they do is the Gambler's Fallacy.
After 3 sixes in a row on a die, P(6) on the next roll is still 1/6. A coin or die has no memory. Probability predicts the long run, not the next toss.
Notes
Trials are independent: a coin or die has no memory, so past results never change the next probability. Believing otherwise is the Gambler's Fallacy.
Check yourself
A fair coin shows heads 6 times in a row. What is the probability of tails on the next toss?
You rolled three 6s in a row. You think "I cannot get a 6 again!" This thinking is called…
A fair die has just shown 6, 6, 6. What is the probability of a 6 on the next roll? Give a fraction as a decimal, rounded to two decimal places.
Answer: 0.17
P(6) is still 1/6, which is 0.1666… ≈ 0.17.
What can probability tell us?
- 1/2 — correct. Yes. The coin has no memory, so the chance is still 1/2.
- More than 1/2. This is the Gambler's Fallacy. The coin does not make up for earlier tosses.
- Less than 1/2. The earlier heads do not change the next toss.
- The Law of Large Numbers. That law is about many trials, not about the next one.
- The Gambler's Fallacy — correct. Yes. Each roll is independent.
- A theoretical probability. It is a mistaken belief about chance.
- Exactly what the next toss will be. In a random experiment the next result cannot be predicted.
- Which side the coin will favour next. A fair coin favours no side.
- What tends to happen in the long run — correct. Yes. It does not tell us what the next result will be.