Skip to content
kuazaraKuazara
Log in

Curiosity · Chance and probability

The gambler's fallacy

If a coin has landed heads five times in a row, is tails now more likely? Find out why a coin has no memory, how gamblers in Monte Carlo lost fortunes to this fallacy in 1913, and what independent events really mean.

  • 14 min
  • 2 simulations
  • Mini quiz

Updated:

Preview

The question

A coin has landed heads five times in a row. If you had to bet on the sixth toss, which would you pick? Many people think "it's tails' turn now". Others say "the heads streak will continue". Which of them is right? Maybe neither.

Try it yourself

No account needed: change the values and check your answer. Some simulations ask for a prediction before they start; it doesn't matter whether it's right or wrong.

Probability lab

After five heads

Toss the coin 100 and 1,000 times. In the after-streak panel, see how many of the tosses that followed five heads were tails. These tosses are rare, so you may need a few thousand tosses.

Prediction

Predict first

The coin has landed heads five times in a row. What do you think the probability of tails is on the sixth toss?

It doesn’t matter whether you’re right; the simulation opens once you choose.

Probability lab

The die that won't roll a 6

Roll the die 1,000 times at a go and watch the after-streak panel. It counts how many times a 6 came up on the roll straight after eight rolls without a 6. Compare this with the proportion of 6s over all rolls.

Prediction

Predict first

The die hasn't rolled a 6 eight times in a row. How does the probability of a 6 on the ninth roll change?

It doesn’t matter whether you’re right; the simulation opens once you choose.

Share: Share to Google Classroom (opens in a new tab)

Embed the simulation on your site

Paste this code into the HTML of your blog or school site. Visitors try the simulation without an account; no progress is saved.

The embedded page sets no cookies and does no tracking. A small link back to Kuazara sits underneath.

Log in to continue

The rest of the lesson is waiting for you, with a mini quiz. Creating an account is free.

Start for freeLog in

Frequently asked

5 questions

If a coin has landed tails five times, isn't heads more likely on the sixth toss?

No. A coin has no memory; every toss starts fresh, knowing nothing of the earlier ones. Because the tosses are independent, the chance of heads on the sixth toss is still 1/2.

Isn't five tails in a row very rare anyway?

Looked at in advance, yes: its probability is 1/32. But those five tails have already happened. From here, the sequences "six tails" and "five tails then one heads" each have probability 1/64; they are equal.

Doesn't this contradict the law of large numbers?

No. The law works by dilution, not by making up for the past. Five extra tails become unimportant within thousands of tosses, which is why the proportion approaches 0.5. The coin never "pays back" anything.

When do past results really matter?

When the events are dependent. If you draw without putting items back, each draw changes what is left. If you take two red marbles without replacement from a bag of 3 red and 3 blue, the chance that the third is red is now 1/4.

Why are long streaks in truly random sequences so surprising?

People expect random sequences to always look mixed up. In truly random sequences, though, long streaks turn up far more often than in sequences people make up; seeing a streak of 5 or more identical results in 100 tosses is quite normal.