How Coin Flips Work: Probability, Math, and Common Myths

The coin flip is the simplest random event most people encounter. Two outcomes, equal odds, no skill involved. Yet this basic mechanism sits at the heart of probability theory, decision-making, and some of the most persistent misconceptions about randomness. Understanding how a coin flip actually works—mathematically—gives you a foundation for thinking about risk, chance, and fairness in far more complex situations.

This guide walks through the probability math, explains why your intuition about randomness is often wrong, and covers real-world uses of coin flips. Want to flip a coin right now? Try our Free Online Coin Flipper.

Fair Coin Probability: The Basics

A fair coin has two equally likely outcomes. The probability of each:

$P(Heads) = P(Tails) = \frac{1}{2} = 0.5 = 50\%$

This means that on any single flip, you have exactly a 50% chance of heads and a 50% chance of tails. The previous flip, the next flip, and every other flip are independent events—they have no influence on each other.

Multiple Flips

When you flip a coin multiple times, the probability of a specific sequence is calculated by multiplying the individual probabilities:

$P(sequence) = \left(\frac{1}{2}\right)^n$

Where n is the number of flips.

Example: What is the probability of flipping 3 heads in a row?

P = (1/2)^3 = 1/8 = 12.5%

Example: What is the probability of flipping 10 heads in a row?

P = (1/2)^10 = 1/1024 = 0.098%

A critical point: the probability of flipping HHHHHHHHHH (ten heads) is exactly the same as flipping HTHTTHTTHH or any other specific sequence of ten flips. Both are 1/1024. We perceive the ten-heads sequence as “special” only because we grouped the outcomes into a pattern. The coin doesn’t know or care about patterns.

Counting Outcomes

With n flips, the total number of possible sequences is 2^n:

Flips	Possible Sequences	Example
1	2	H, T
2	4	HH, HT, TH, TT
3	8	HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
5	32
10	1,024
20	1,048,576

With just 20 flips, there are over one million possible sequences. This gives you a sense of how quickly combinatorial possibilities expand.

The Binomial Distribution

When you flip a coin n times and count the number of heads, the result follows a binomial distribution. This tells you the probability of getting exactly k heads in n flips:

$P(k) = \binom{n}{k} \times p^k \times (1-p)^{n-k}$

Where the binomial coefficient is:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Example: What is the probability of getting exactly 3 heads in 5 flips?

Binomial coefficient: 5! / (3! x 2!) = 10
P(3) = 10 x (0.5)^3 x (0.5)^2 = 10 x 0.125 x 0.25 = 31.25%

Full distribution for 5 flips:

Heads	Probability
0	3.125%
1	15.625%
2	31.25%
3	31.25%
4	15.625%
5	3.125%

Notice the symmetry and the bell shape. Getting 2 or 3 heads (the middle outcomes) accounts for 62.5% of all results. Getting 0 or 5 heads (the extremes) accounts for only 6.25%.

With 100 flips: Getting exactly 50 heads happens only about 8% of the time. Getting between 45 and 55 heads happens about 73% of the time. The distribution gets taller and narrower (relative to the range) as n increases—outcomes concentrate around the expected value.

The Law of Large Numbers

The law of large numbers states that as you increase the number of trials, the observed proportion of outcomes will converge toward the theoretical probability. Flip a coin 10 times and you might get 70% heads. Flip it 10,000 times and you will almost certainly be within 1-2% of 50%.

Worked example of convergence:

Flips	Heads	Proportion
10	7	70%
100	53	53%
1,000	512	51.2%
10,000	5,023	50.23%
100,000	50,084	50.084%

The proportion trends toward 50%, but it does so by dilution—not by correction. After 10 flips showing 7 heads, the coin does not “owe” you extra tails. Instead, the next 10,000 flips will be roughly 50/50, and those 3 extra heads from the first batch become statistically insignificant in the larger sample.

This distinction matters enormously and leads directly to the most famous probability misconception.

The Gambler’s Fallacy

The gambler’s fallacy is the belief that past random outcomes influence future ones. If a roulette wheel has landed on red five times in a row, many gamblers will bet on black because they feel it’s “due.” This is wrong.

Why it feels right: Your brain expects random sequences to look balanced in the short run. Five heads in a row seems unbalanced, so your intuition says tails should come next to even things out. But the coin doesn’t have memory. Each flip is a fresh 50/50 event regardless of what came before.

The math: After five heads in a row, the probability of the next flip being heads is still exactly 50%. The probability of the next flip being tails is still exactly 50%. The sequence HHHHHH and HHHHHT are equally likely from this point forward.

Where people go wrong: They confuse two different questions:

“What is the probability of flipping 6 heads in a row?” Answer: (1/2)^6 = 1.56%. This is low.
“Given that I have already flipped 5 heads, what is the probability the 6th flip is heads?” Answer: 50%. The first 5 flips already happened—they are no longer uncertain.

The gambler’s fallacy has cost people enormous sums. On August 18, 1913, at the Monte Carlo Casino, the roulette ball landed on black 26 times in a row. Gamblers lost millions betting on red, convinced that the streak had to end. Each spin was independent. The table didn’t care about the streak.

Are Real Coins Fair?

In theory, a coin is a perfect 50/50 device. In practice, physical coins have slight biases.

The Diaconis study (Stanford, 2007): Mathematician Persi Diaconis and colleagues built a coin-flipping machine and found that a coin is approximately 51% likely to land on the same side it was facing when flipped. This is because the coin spends slightly more time in the air with its starting face up due to precession (wobble). The bias is tiny—51/49—and undetectable over a small number of flips.

Other physical factors:

Coin construction: Some coins are not perfectly balanced. A US nickel, for example, has different designs on each side with slightly different weight distributions
Flipping technique: A coin that spins (like a quarter on a table) behaves differently than one that tumbles through the air
Catching vs. landing: Catching a coin in your hand adds randomness compared to letting it hit a flat surface

For practical decision-making, these biases are negligible. A coin flip is as close to a fair 50/50 outcome as you can get without a computer. For scientific or statistical purposes where exact fairness matters, use a digital random number generator—like our Coin Flipper.

Coin Flips in Decision-Making

Using a coin flip to make decisions is more useful than it first appears, and not always for the reason you’d think.

The direct method: When two options are genuinely equal in expected value, a coin flip prevents analysis paralysis. Should you eat Italian or Thai for dinner? Flip a coin. The time you save not deliberating is worth more than the marginal difference between the two options.

The reveal method: Flip a coin and pay attention to your emotional reaction the instant it lands. If you feel relief, the coin chose what you actually wanted. If you feel disappointment, go with the other option. The coin flip acts as a tool to surface your hidden preference, not as the actual decision-maker.

Research support: A 2020 study by economist Steven Levitt (of Freakonomics fame) asked people facing major life decisions to flip a coin. Those who made a change (quit a job, ended a relationship, moved cities) reported being happier six months later than those who maintained the status quo. The coin served as permission to act on a decision they had already been leaning toward.

Historical Coin Flip Moments

The Wright Brothers (1903): Wilbur and Orville Wright flipped a coin to decide who would pilot the first powered flight at Kitty Hawk. Wilbur won the toss but stalled on his attempt on December 14. Orville flew successfully three days later on December 17. The coin determined who tried first, but skill and conditions determined who succeeded.

The NFL Overtime Rule: Professional football has used a coin flip to determine possession in overtime since 1974. The rule was modified in 2010 (and again in 2024) because statistical analysis showed that the team winning the toss won the game roughly 60% of the time—the advantage of receiving the ball first was significant enough to make the coin flip feel unfair despite being perfectly random.

Portland Trail Blazers (1970): The NBA held a coin flip between the Portland Trail Blazers and the Milwaukee Bucks for the first pick in the 1970 draft. Milwaukee won and selected Kareem Abdul-Jabbar (then Lew Alcindor), who became one of the greatest players in basketball history. A single coin flip shaped two franchises for a decade.

Frequently Asked Questions

If I flip a coin 100 times and get 60 heads, is the coin biased?

Not necessarily. Getting 60 heads in 100 flips has about a 2.8% probability with a fair coin. That’s unusual but not astronomically rare—it would happen roughly once in every 36 sets of 100 flips. To establish genuine bias, you would need hundreds or thousands of flips. Statisticians use hypothesis testing (specifically a binomial test) to determine whether an observed result is significantly different from 50/50.

What is the expected number of flips to get a specific sequence like HTH?

This is trickier than it seems. The expected number of flips to see the sequence HTH is 10, while HHH requires only 8 on average, despite both being three-flip sequences. The difference arises because partial matches can overlap: after failing to complete HHH (getting HHT), you’re starting completely over, but after failing HTH (getting HTH… wait, that’s already a match context issue). The math involves Markov chains and varies by sequence.

Can a coin land on its edge?

Technically yes, but the probability is extremely small. Physicist Daniel Murray estimated the odds at approximately 1 in 6,000 for a US nickel spun on a table (nickels have relatively thick edges). For a coin flipped in the air and caught, the probability is effectively zero. Most game rules treat an edge landing as a re-flip.

Why do sports use coin flips instead of other random methods?

Coin flips are fast, transparent, and universally understood. Everyone can see the coin go up and come down. There’s no equipment to malfunction, no software to question, and no complicated procedure to explain. The fairness is visually obvious in a way that pulling a number from a hat or using a random number generator is not, even though all methods are equally random.

How many flips would I need to be 95% confident the coin is fair?

Using a standard binomial test with a significance level of 0.05, you would need approximately 1,000 flips to reliably detect a small bias (like 52/48). For a larger bias (like 55/45), around 200 flips would suffice. For the tiny 51/49 bias found in the Diaconis study, you would need tens of thousands of flips to confirm it statistically. The smaller the bias, the more data you need.