Streaks and the gambler's fallacy

What you will be able to do

• explain the gambler's fallacy and why it is a mistake for independent events;
• show that every sequence of fair tosses of the same length is equally likely;
• estimate by simulation how long the streaks in a random sequence usually are.

Tails must be due

A fair coin lands heads $5$ times in a row. Is tails now more likely than heads?

No. Each toss is independent of the ones before. The coin has no memory of the last five tosses, so the next toss is still heads with probability $\dfrac{1}{2}$. The feeling that a result is "due" after a streak is called the gambler's fallacy. A fallacy is a mistake in reasoning that seems right.

A streak looks special, but it is not. Write a particular sequence of $6$ tosses, such as HTTHTH. Its probability is $\left(\dfrac{1}{2}\right)^6 = \dfrac{1}{64}$. The sequence HHHHHH has the same probability, $\dfrac{1}{64}$. It only looks less likely because there are many sequences that look mixed, and only two that look like this, all heads or all tails.

Streaks are normal

How long is the longest streak in $100$ tosses of a fair coin? Many people guess $3$ or $4$. Write your own guess down, then run the program.

A long streak in a real sequence is not a sign that the coin is unfair, or that the next result is due. If you try to write $100$ "random" tosses by hand, you will probably switch between H and T too often. A real coin gives longer runs than most people expect.

When the past does matter

The gambler's fallacy is a mistake only when the tries are independent. When cards are dealt without replacement, the past changes what is left. If $20$ cards from a pack have been dealt with no heart among them, the remaining $32$ cards hold all $13$ hearts. Then a heart really is more likely than at the start: $\dfrac{13}{32} \approx 0.41$ against $\dfrac{13}{52} = 0.25$.

So ask one question before you trust a streak: does the past change the chances of the next try?

Your turn

The usual mistakes

• "It is due." For independent tries, the next result has the same chance as ever.
• "It is hot." The opposite mistake: after a streak, expecting it to go on. For a fair coin this is just as wrong.
• Calling a streak proof of a fault. Long runs are normal in long random sequences.
• Forgetting the exceptions. Draws without replacement do depend on the past.

Remember these

• A fair coin or die has no memory: the past does not change the next try.
• Every particular sequence of $n$ fair tosses has probability $\left(\dfrac{1}{2}\right)^n$.
• In $100$ fair tosses, a run of $5$ or more equal faces happens about $97$ times in $100$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A fair coin has landed tails $7$ times in a row. What is the probability of heads on the next toss?

2. Which is more likely in $4$ tosses of a fair coin: HHHH or HTHT?

3. A fair die has not shown a six in $12$ rolls. Find the probability that the next two rolls are both sixes.

4. Cards are dealt from one pack without being put back. After $26$ cards with no ace, what is the chance that the next card is an ace?

5. A friend says a coin must be biased because it gave $6$ heads in a row somewhere in $100$ tosses. What do you say?

References

• OpenStax, Introductory Statistics, Chapter 3, the section on independent events.