Can you fake a coin? Real coins streak more than you think
Flip a fair coin 100 times and you will probably get a long streak of the same face. A run of six or more in a row turns up 80.7% of the time, and the longest run is 7 or more in 54.2% of sequences. People making up a “random” sequence tend to leave the long runs out, as the statistician Mark Schilling has noted — which is how a fake gets caught.
Try it
Type the sequence you think a real coin would give. 0 of 50.
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How long the longest run is
Chance that the longest run of one face in 50 fair flips is exactly this long — the length you chose above.
| Flips | Typical longest run | 5 or more | 6 or more | 7 or more |
|---|---|---|---|---|
| 20 | 4 | 45.8% | 23.7% | 11.5% |
| 50 | 6 | 82.1% | 54.4% | 30.9% |
| 100 | 7 | 97.2% | 80.7% | 54.2% |
| 200 | 8 | 99.92% | 96.5% | 79.9% |
“Typical” is the median: half of all sequences reach it.
Why the runs get so long
Every time the coin changes face, a new run starts, and that happens on about half the flips. Each run then has an even chance of lasting one more flip, so you can count down. For 50 flips, roughly:
- Runs in allabout 26
- Runs at least 2 longabout 13
- Runs at least 3 longabout 6
- Runs at least 4 longabout 3
- Runs at least 5 longabout 2
- Runs at least 6 longabout 1
Somewhere around a run of 6 you are down to about one, and that is the typical longest run — the exact figure for 50 flips is 6. The same countdown lands within one of the exact typical longest run at every length in the table.
The halving also says why more flips barely help: twice as many runs means just one more halving before you run out, so doubling the flips should add about one to the longest run. The exact figures agree: the typical longest run goes from 6 at 50 flips to 7 at 100 and 8 at 200.
How the exact figures are worked out
The halving is a rough guide; the table is exact. It counts, for every length of run, how many of the 2n possible sequences never contain a run that long, by building the sequences one flip at a time and keeping track of how long the current run is. A sequence with a run of k of either face is the same thing as k − 1 “same as the last flip” marks in a row among the n − 1 gaps between flips, which is the shortcut Schilling uses; the generator checks that shortcut against a direct count.
Questions people ask
- What is the longest streak of heads or tails in 100 coin flips?
- Usually about 7: 54.2% of sequences of 100 fair flips have a run of 7 or more of the same face. A run of 6 or more turns up 80.7% of the time.
- How can you tell a fake coin-flip sequence?
- Faked sequences usually leave out long runs, which also makes them switch between heads and tails too often. In 50 real flips the longest run is 6 or more half the time, and there is a run of at least 5 in 82.1% of sequences.
Check it yourself
Every figure here is an exact count over all possible sequences, checked against published figures before it ships.
Long Run Predictions — Mark F. Schilling, Math Horizons, Spring 1994
The readable one, from a statistician who has written on exactly this. It gives figures for 250 tosses; this page computes the same thing for 20 to 200 flips, and checks itself against his: a run of at least six heads in 87% of sequences, seven in 63%, eight in 38%, and a run of eight of either face in 63%. All four are reproduced here exactly. Schilling also observes that people asked to write a random-looking sequence are reluctant to include more than four or five of a kind in a row.
This page’s dataset — Odds The Law
The full distribution of the longest run and of the number of switches for 20, 50, 100 and 200 flips.
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How the site works out its numbers is on the methodology page.