A single toss and a short run
A fair coin lands heads with probability 1/2 and tails with probability 1/2. Each toss is independent of the ones before. The chance of heads twice in a row is 1/2 x 1/2 = 1/4, three in a row is 1/8, and five in a row is 1/32, about 3.1 percent.
Those numbers answer one particular question: starting from the next toss, what is the chance that the next k tosses are all heads? That is not the same as asking how likely a run of k heads is to show up somewhere in a longer sequence. The second question has a much bigger answer, and the difference explains why streaks surprise people.
Why streaks appear in longer sequences
In a sequence of 20 tosses, there are many places where a run of five heads could begin: starting at toss 1, toss 2 and so on, up to toss 16. Each start has a small chance, but there are many chances. The probabilities do not add exactly, because the starting points overlap, but the effect is real.
The exact figures can be built by counting sequences that do not contain a run of k heads. For runs of two heads, those counts follow the Fibonacci pattern: 2, 3, 5, 8, 13 for sequences of 1 to 5 tosses. In 4 tosses, 8 of the 16 sequences contain no pair of consecutive heads, so the chance of at least one HH is 1 - 8/16 = 1/2.
Some computed chances
The table gives the chance that a sequence of n fair tosses contains at least one run of k or more heads. Values come from the counting method above and are rounded.
| Tosses | Run length | Chance of at least one run |
|---|---|---|
| 4 | 2 | 50% |
| 10 | 2 | 86% |
| 10 | 3 | 51% |
| 20 | 5 | 25% |
| 100 | 5 | 81% |
| 100 | 6 | 55% |
| 100 | 7 | 32% |
A run of three heads in 10 tosses is about a coin-flip event. A run of seven heads somewhere in 100 tosses happens roughly one time in three. If you ask a group to write down a random-looking sequence of 100 tosses, they often avoid runs this long, which is one way people can tell real random sequences from invented ones. Real randomness is clumpier than intuition suggests, a theme that also runs through random streaks in other settings.
How long should the longest run be?
As a rough guide, in n fair tosses the longest run of heads is typically around log base 2 of n. For 100 tosses, that suggests about 6 or 7, which agrees with the table, where runs of 6 or 7 are common but not guaranteed. This is an approximation, not an exact rule, and the actual longest run varies from sequence to sequence.
A related fact: the expected number of tosses needed to see the first run of k heads is 2^(k+1) - 2. For two heads (HH), that is 6 tosses. For three, it is 14. For five, it is 62. These are averages across many repeated experiments, and any single experiment may be far shorter or longer.
A streak does not change the next toss
The most important point is what streaks do not mean. After four heads in a row, the next toss of a fair coin is still 50-50. The chance of five heads in a row, viewed from the start, is 1/32, but once four have already happened the question is only about the next toss. This is the gambler’s fallacy in its classic form: believing tails is “due”.
The opposite mistake is also common. Some people see a streak and conclude the coin must be biased or that the “hot hand” is real. A streak in a short sample can be completely ordinary. Whether a coin is fair is better judged from the total counts over many tosses, with a sense of how much small samples vary. Even a long run is weak evidence on its own, although an extremely long run (say 30 heads) would be reason to doubt the assumption of a fair coin.
Clustering and the law of averages
Another myth is that a streak will be “corrected” by a run of tails. The count of heads minus tails can drift far from zero and does not have to return quickly. What does happen is that the proportion of heads tends toward one half as the number of tosses grows. The difference between those two ideas is the gap behind the law of large numbers, and why the so-called law of averages is a misleading name. A sequence can keep an excess of heads, in raw numbers, and still see the proportion approach 50 percent.
Notes on how to use these numbers
These figures are for illustration of probability and apply only to a fair coin with independent tosses. They do not give a way to predict the next toss or to gain an advantage in any game of chance. In games with a house edge, the long-run advantage stays with the operator regardless of past streaks. If gambling is causing harm, a local support service can help.