Streak Odds: Chance of a Run in a Row
A run of five in a row feels rare, yet in 100 rounds it is more likely than not. Set the chance of each round, the number of rounds and the run, and get exact answers that count every place a run could start.
Two questions that sound the same
“What are the chances of five heads in a row?” can mean two things. The first is about five rounds picked in advance. All five must go the same way, which at 50% a round is 1 in 32. The second is about a whole stretch: will five in a row appear somewhere in 100 rounds? That chance is 81%, because a run can start at any of 96 places.
People feel the first number while watching the second one happen, which is why a streak seems to mean something. The calculator keeps the two apart and answers the second.
How to use it
Type the chance that each round goes your way (50 for a coin, 40 for a colour that covers four of ten digits, 10 for one number), the number of rounds in the stretch, and the length of the run. Your outcome is the one whose chance you typed. Choose “your outcome or its opposite” when either side counts, as with big or small.
The tiles show the chance of at least one run, how many runs to expect on average, the longest run half of all stretches reach, the chance for rounds picked in advance, and how long the wait is. The table repeats the stretch for other run lengths. The four starting points are illustrations, so replace them with the chances your own app states.
A worked example: five heads in 100 flips
| Run of | Chance in 100 flips |
|---|---|
| 3 | over 99.9% |
| 4 | 97% |
| 5 | 81% |
| 6 | 55% |
| 8 | 17% |
| 10 | 4.4% |
| 12 | 1.1% |
The longest run to expect in 100 flips is 6: half of all stretches reach it. On average 1.52 runs of five or more appear. The first run of five takes 62 flips on average and is more likely than not by flip 44. Every one of these figures is exact.
The same idea for a colour, a number and big or small
- A colour with a 40% chance. Four in a row in 60 rounds turns up in 61% of stretches. For four rounds picked in advance it is 1 in 39.
- One number with a 10% chance. Three in a row in 1,000 rounds turns up in 59% of stretches, while three picked rounds all matching is 1 in 1,000. The first such run takes 1,110 rounds on average.
- Big or small, either side, 1,440 rounds. That is one round a minute for a day. A run of 8 alike is almost certain, the longest run to expect is 11, a run of 10 appears in 76% of days and a run of 12 in 30%.
How the numbers are worked out
There is no guessing and no simulation behind the tiles. The calculator counts every way a stretch can go. For one outcome, the chance that a run first finishes at round j is the chance of no run in the first j − k − 1 rounds, times one miss, times k hits. For either side it follows the current side and the length of its run round by round. The average number of runs is p^k × (1 + (n − k) × (1 − p)), and the average wait is (1 − p^k) ÷ ((1 − p) × p^k), which for a coin is 2^(k+1) − 2.
The round by which a run is more likely than not is found exactly. Where it passes 2,00,000 rounds it is taken from 0.69 times the average wait, marked “about”, and stays within one part in ten thousand. The box at the bottom draws 1,000 random stretches with pretend rounds: about 810 of them hold a run of five, as the exact figure says.
What a streak does not tell you
Rounds here have no memory. After five heads the next flip is still 50%. A run is not a trend, so the same side is not likelier next, and it is not a debt, so the other side is not due. Independent rounds do neither.
What the calculator gives you is how often runs happen, so that one seen on a chart is not mistaken for a signal. It cannot tell you whether an app’s rounds are independent, because we cannot see inside any app, and every chance in it is one you type. See the colour history analyser to check a pasted list, the streaks page of the prediction simulator for random rounds drawn live, and colour tricks and formulas for why chart patterns do not predict. The Win Go guide explains how a round is played, and the free tools page lists the rest.
Common questions
What are the odds of getting five in a row?
For five rounds picked in advance at 50% each, it is 1 in 32. For five in a row somewhere in 100 rounds it is 81%, because the run can start at any of 96 places. The calculator shows both so you can see the difference.
After five reds in a row, is green more likely next?
No. In independent rounds the next round has the same chance whatever came before. A run does not make the other colour due, and it does not make the same colour likelier either. The calculator treats every round as a fresh draw.
How many rounds before a streak of eight is likely?
For big or small at 50%, counting either side, a run of eight alike is more likely than not by round 179 and takes 255 rounds on average. Change the chance or the run length and the calculator gives the new figures.
Why does a long streak turn up so often when I watch?
A streak can start at any round, so a long stretch gives it many chances. In 1,440 rounds of big or small, a run of 8 alike is almost certain, and the longest run to expect is 11.
Is this the chance of winning several rounds in a row?
The arithmetic is the same if you type the chance of winning a round. It says nothing about any app’s real chances, which you must take from the app itself. Several wins in a row is ordinary luck, not a sign of skill or of a pattern.
Do the numbers I type here leave my phone?
No. The chances are worked out in your browser. If you press the copy-link button, the numbers go into the link itself, after the # sign, which a website never receives.