About the Coin Flip Simulator
Flipping a coin is the simplest way to make a fair decision, and one of the classic examples in probability. With a fair coin, heads and tails are equally likely every time. But flip a coin many times and the results can surprise you: long streaks, uneven totals, and rarely exactly half heads. Seeing this happen is one of the best ways to understand randomness.
This coin flip simulator flips a virtual coin once or up to 10,000 times. It shows the result or the count of heads and tails, the sequence of flips, the share of heads, how many heads to expect, how likely your exact result was, and the longest streak.
How to Use the Coin Flip Simulator
Enter the number of flips: 1 for a single decision, or more to see how the results spread.
Press Calculate to flip, and again to flip again.
Enter a seed if you want the same flips every time, for example to share a result.
How the Probabilities Are Worked Out
expected heads = flips ÷ 2
typical spread = ± √flips ÷ 2 (one standard deviation)
chance of exactly k = C(flips, k) ÷ 2^flipsStep-by-Step Example
100 flips.
Expected heads: 100 ÷ 2 = 50
Typical spread: √100 ÷ 2 = ±5
Chance of exactly 50: C(100, 50) ÷ 2^100 = about 8%10 flips.
Chance of exactly 5 heads: 252 ÷ 1,024 = 24.6%Why You Rarely Get Exactly Half
Even though 50 heads is the single most likely result from 100 flips, it happens only about 8 percent of the time. Results between 45 and 55 heads cover most flips, and anything from 40 to 60 is not unusual. As the number of flips grows, the share of heads gets closer and closer to 50 percent — this is the law of large numbers — but the difference in the count of heads and tails usually grows, just more slowly than the number of flips.
Streaks Are Normal
People expect random sequences to alternate much more than they really do. In 100 flips, the longest run of the same side is usually 6 or 7, and runs of 8 or more are common. When people invent a "random" sequence, they rarely include runs longer than 4. This is why a long streak in a real game or investment is not, on its own, evidence that something is wrong.
The Gambler's Fallacy
After five heads in a row, many people feel tails is "due". It is not. The coin has no memory, and each flip is still 50–50. The belief that past results change future odds is called the gambler's fallacy, and it has cost gamblers dearly for centuries. The opposite belief — that a streak will continue — is also a fallacy for a fair coin.
Are Real Coins Fair?
Research has found that a coin flipped by hand is very slightly more likely to land the same way up as it started, by about 1 percent, because of the way coins wobble in the air. Coins spun on a table can be far less fair, because the weight of the design makes one side more likely. For practical decisions these effects are negligible, and virtual coins avoid them entirely.
Using a Coin Flip to Decide
A coin flip is a fair way to settle a choice between two options, choose who goes first or break a tie. Agree which side means what before flipping. Some people find a coin flip useful even when they do not follow it: noticing whether you hoped for heads or tails can reveal what you really want.
Coin Flips in Probability Lessons
Coin flips are the starting point for much of probability theory. Counting the ways to get a given number of heads leads directly to Pascal's triangle and binomial coefficients: with 4 flips there is 1 way to get no heads, 4 ways to get one, 6 ways to get two, 4 ways to get three and 1 way to get four, out of 16 equally likely sequences. The same pattern, scaled up, describes everything from quality control in factories to the spread of results in opinion polls. A classic classroom exercise is to have each student flip a coin 20 times, then combine the class's results: individual results vary widely, but the combined share of heads comes very close to one half.
Heads or Tails for Sport
Coin tosses decide who kicks off in football, who bats first in cricket and who receives first in American football. In each case, the toss gives both sides an equal chance of the advantage, and the captain who wins the toss chooses. Some competitions have replaced a physical toss with an electronic one to avoid any doubt about fairness.
Understanding Your Result
Result gives the side for one flip, or the count of heads and tails.
Sequence shows the flips in order, up to the first 100.
Share of heads gives the percentage of heads.
Expected gives the expected number of heads and the typical spread.
Chance of this result shows how likely your exact count was.
Longest streak shows the longest run of one side.
When Should You Use This Calculator?
Making a quick fair decision.
Choosing who goes first in a game or match.
Teaching probability and the law of large numbers.
Exploring streaks and randomness.
Settling a friendly bet.
Common Mistakes
Believing a result is "due" after a streak.
Expecting exactly half heads.
Treating long streaks as proof of an unfair coin.
Deciding what each side means after the flip.
Using a seed when you want a fresh flip.