After a run of heads, is the next flip more likely to be tails?

No. Coin flips are independent. After any run of heads, the next flip is still 50% tails. Believing otherwise is the gambler's fallacy.

Gambler's fallacy · "we're due for a change"

Five heads in a row. Tails is due — right?

Flip a fair coin thousands of times. Every time k of the same face land in a row, we tally what comes next. The intuition says the streak must break. The coin has no memory.

After a run of 5 heads, the share of next flips that are tails:

— %

theory 50.0% · 0 runs observed

Flip tape · streaks of 5+ highlighted

H T run end

Heads

—

0 flips

Tails

—

0 flips

Longest run

expected ≈ log₂ N

Flip the coin

Streak length to test

After a run of this many in a row5

Which face starts the streak

"After streak" rate vs flips0 runs

next is opposite 50%

How to use

Hit +10,000. The sim flips a fair coin, draws the tape on the left, and scans for every run of 5 in a row. For each run it records the very next flip. The big number up top is the share that broke the streak — it lands on 50%, exactly the unconditional rate. The streak predicts nothing.

The fallacy

The gambler's fallacy is the belief that past random events influence future independent ones. After 7 heads, "we're due for tails." We aren't. The 8th flip is still 50/50.

It feels wrong because we expect samples to look like the population. We unconsciously imagine the coin has memory, or "evens out." Independent randomness has no such mechanism. Each flip is fresh.

The fallacy works in reverse too — the hot hand fallacy: "she's on a streak, she'll keep going." Also nonsense for truly random processes.

Run lots of flips to verify

Flip 10,000 times. Then look at every streak of 5+ heads. Count what came next.

The "after streak" tails fraction will converge to 50% ± noise, identical to the unconditional rate. The streak doesn't predict the next flip.

Notice though: with 10,000 flips, you'll see streaks of 13+ heads. That feels astronomical but is mathematically guaranteed — expected longest run of either side is ~log₂(N) ≈ 13.

The Monte Carlo Casino, 1913

On 18 August 1913, roulette at the Monte Carlo Casino landed on black 26 times in a row. Gamblers piled increasingly large bets on red, certain it was "due." Each spin was independent. The casino made millions.

The probability of 26 blacks in a row: about 1 in 67 million. Rare, but it had to happen eventually somewhere — and the gamblers' losses were not the rarity. Their certainty about the next spin was the fallacy.

When streak intuition is RIGHT

Not all streaks are independent. In contexts where there's a hidden state (a player's confidence, a hot stove, a biased coin), past events do predict future ones. The hot hand in basketball, for example, has been re-examined and shows small but real effects.

The fallacy is specifically about provably independent processes — coin flips, roulette, lottery numbers, dice. Whenever there's no causal link between outcomes, history tells you nothing about the future.

Citations

Tversky, A. & Kahneman, D. (1971). "Belief in the law of small numbers." Psychological Bulletin 76(2): 105–110. The classic study introducing what would later be called the gambler's fallacy.
Croson, R. & Sundali, J. (2005). "The gambler's fallacy and the hot hand: Empirical data from casinos." Journal of Risk and Uncertainty 30(3): 195–209. Roulette-table evidence of bias-against-streaks betting.
Miller, J.B. & Sanjurjo, A. (2018). "Surprised by the hot hand fallacy? A truth in the law of small numbers." Econometrica 86(6): 2019–2047. Re-examined the original Gilovich–Vallone–Tversky hot-hand result and found a small but real effect.
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1 (3rd ed.). Wiley. Ch. III on run lengths in Bernoulli sequences.

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