Coin Flip Probability: Odds, Streaks and Worked Examples

On a fair coin, the probability of heads is 1/2, or 50%, and the probability of tails is the same. That figure describes one flip. The harder questions are about several flips: how likely is a streak, how many heads should you expect in 10 or 100 tosses, and does a run of tails make heads “due”? This article answers each one with the arithmetic shown, so you can check every number yourself or run it in a coin flip simulator.

TL;DR

• One fair flip: P(heads) = P(tails) = 1/2 = 50%. The U.S. Mint’s educational guide gives the same figure.
• Flips are independent. Earlier results never change the next flip, so a streak does not make the opposite side overdue.
• One specific ordered sequence of n flips has probability (1/2)^n: seven tails in a row is 1/128 (about 0.78%), ten in a row is 1/1,024 (about 0.098%).
• Exactly k heads in n flips is C(n,k) / 2^n. Ten flips give exactly five heads only about 24.6% of the time.
• Real coins are slightly imperfect, but the measured departure from 50/50 is small.

Table of Contents

• The odds on a single flip
• Why the last flip does not change the next one
• Probability of a specific sequence
• Streaks somewhere in a long sequence
• How many heads to expect in n flips
• Does a real coin land 50/50?
• Combining coin flips with other events
• Using a coin flip as a decision tool
• Run the experiment yourself
• Common mistakes to avoid
• FAQ
• Sources

The odds on a single flip

Probability is the number of favorable outcomes divided by the number of equally likely outcomes. A coin has two outcomes, heads and tails, so each has probability 1/2. The U.S. Mint’s guide states it this way: two possible outcomes, so heads has a probability of 1/2, or 50%. A university course from Waterloo reasons the same way: when the two outcomes are equally likely, tails is 1/2, and by symmetry so is heads.

The same number can be written several ways, which is useful when you meet different phrasing:

The last row matters if you search for “coin toss odds”. In everyday speech, odds and probability get mixed up. Probability compares the favorable outcome to all outcomes (1 of 2). Odds compare favorable to unfavorable (1 to 1). For a fair coin both describe the same even split, but for other events the two numbers differ, so keep the distinction in mind. To convert between fractions, decimals and percentages, the fraction calculator and the percentage calculator handle the arithmetic exactly.

What “fair” assumes

Every figure in this article rests on a model: a fair coin, independent flips, two outcomes. Wolfram MathWorld separates this ideal model from physical coins, which have thickness and side-dependent effects. The model is a very good approximation, not a guarantee about a particular coin on a particular table. We come back to the measured size of that gap below.

Why the last flip does not change the next one

The most common mistake in coin probability is the belief that a streak must correct itself. After five heads, many people feel that tails is more likely. It is not. An ANU mathematician explains that each flip is independent of the last, so a run of heads makes neither tails nor another head more likely.

The formal statement is in a lecture from the London School of Economics: independence means P(A and B) = P(A) × P(B), and the next fair flip stays at 1/2 after any earlier sequence. The same lecture connects the gambler’s fallacy to a misreading of independence. A losing streak does not raise the chance of a win when future trials are independent.

Stanford’s course materials treat repeated fair flips the same way, as independent trials with a heads probability of 0.5.

A reconciliation that helps

The confusion usually comes from mixing two different questions:

1. Before any flips, how likely is it that I will get six heads in a row? Very unlikely: (1/2)^6 = 1/64, about 1.56%.
2. I have already flipped five heads. How likely is a sixth? Exactly 1/2.

The first question covers the whole sequence, the second only the next flip. The five earlier heads are now history, not a probability. Probability of the full sequence is small because it requires many independent events to line up, yet the last step of it is still a coin flip.

Probability of a specific sequence

For one particular ordered sequence of n fair flips, multiply the per-flip probabilities. Stanford states the result as (1/2)^n. Here are some values.

Two things are worth stating clearly about this table.

First, the numbers are for a specified run. “Seven tails in a row, starting with my next flip” is 1/128. That is the answer to “how rare is it to flip tails seven times in a row” when you name the starting point in advance.

Second, the same logic shows that HHHHHHH and HTTHTHH are exactly equally likely as ordered sequences, each at 1/128. The first one looks special to us, so we notice it, but the coin has no opinion.

Losing ten flips in a row

If you call heads ten times and lose every time, the chance for a specified ten-flip run is (1/2)^10 = 1/1,024, about 0.098%. Over a longer session, though, the chance of seeing a run like that somewhere is higher, and it depends on how many flips you make. That brings us to streaks in long sequences.

Streaks somewhere in a long sequence

A streak is unsurprising when you give it room to appear. The LSE lecture reports an 83% chance of seeing four consecutive heads somewhere in 50 fair tosses. ANU reports that the average wait for four heads in a row is around 30 flips with one coin.

These two figures show why intuition fails. A specified four-head run starting now is only 1/16 (6.25%). But in 50 flips there are many possible starting positions, and each one is a chance for the pattern to begin. Real random data therefore contains more streaks than most people expect. Fabricated “random” sequences, by contrast, tend to alternate too often.

If you want to see this directly, a simulator that tracks streaks makes the point faster than any formula. The GizmoBench coin flip simulator is built around fair flips, tallies and streaks, so you can run a few hundred flips and check where your longest run lands.

How many heads to expect in n flips

The expected share of heads is 50%, but any short run will wobble around it. For exactly k heads in n flips, count the ways to place the heads and multiply by the probability of each arrangement. Stanford gives this as C(n,k)(1/2)^k(1/2)^(n-k), which equals C(n,k)/2^n for a fair coin. This is the binomial distribution.

Worked example: three flips

There are 2^3 = 8 equally likely ordered outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. The number of heads breaks down as follows.

So “exactly two heads in three flips” is 37.5%, three times likelier than “three heads”, because three different orderings produce it.

Worked example: ten flips

With n = 10 there are 1,024 ordered outcomes. Our calculation from the formula above:

The most likely single result, five heads, still occurs less than a quarter of the time. Adding the central band of four, five or six heads gives (210 + 252 + 210) / 1,024 = 672/1,024, about 65.6%. The U.S. Mint’s own example illustrates the point: its ten-flip run produced 6 heads, or 60%, a perfectly ordinary outcome.

What happens at 100 and 1,000 flips

The Mint also reports 57 heads in 100 flips (57%) and 543 heads in 1,000 flips (54.3%). Two ideas explain why these are normal, and both are our reading of the binomial math rather than claims from that page.

• The count of heads drifts in absolute terms as flips increase. In 100 flips, the standard deviation is 5 heads (the square root of 100 × 0.5 × 0.5). In 1,000 flips it is about 15.8 heads.
• The share of heads narrows toward 50%, because the deviation grows more slowly than the number of flips.

For 100 flips, exactly 50 heads has a probability near 8%, while a count anywhere from 40 to 60 heads covers roughly 96% of outcomes. You can verify these with the binomial distribution calculator, which gives exact probabilities, cumulative values and tails for any n and p.

Richard Feynman puts the long-run idea precisely in his lectures: an honest coin has a heads probability of 0.5, and large numbers of tosses should contain about equal numbers of heads and tails. Note the wording “about equal”. Probability is an estimate of the most likely fraction, not a promise about every finite sequence.

Does a real coin land 50/50?

The ideal model is exact. A physical coin might not be. MathWorld summarizes research on a 2025 study of 350,757 flips: the side facing up at the start landed up again about 50.8% of the time, roughly 51 to 49. The same summary notes that when the starting side was assigned randomly, the data showed no evidence of a heads-versus-tails bias.

Two practical readings follow, and both are our interpretation:

• A same-side tendency of under one percentage point is real in that data but small. For a single call it changes nothing you could feel.
• The bias relates to how the coin starts and is flipped, not to heads being heavier than tails. If you want to remove even that effect, hide which side starts up, or use a digital flip.

MIT’s probability notes make a related point: a coin’s individual motion depends on many variables, yet the pattern across many trials can be described with a simple distribution. That is why a 50/50 model works despite the messy physics.

Combining coin flips with other events

Once a problem has more than one condition, such as “at least one head in two flips” or “heads, and also a six on a die”, the cleanest method is to list the outcomes or use the rules for combining events.

1. Both of two independent events: multiply. Heads and a six on a die: 1/2 × 1/6 = 1/12.
2. At least one: compute the complement. At least one head in 4 flips is 1 − (1/2)^4 = 15/16, or 93.75%.
3. Exactly k of n: use the binomial formula above.

For mixed problems, the probability calculator for two events covers unions, intersections and conditionals with a Venn view, and the dice probability explorer does the same for sums on dice.

Expected value, briefly

The same independence logic is used outside games. A repeated bet with independent outcomes has an expected value per attempt, and the long-run average depends on win rate and payoff, not on the streak you just saw. A trading expectancy formula applies this reasoning to win rates and reward-to-risk ratios, with a worked example of how a win rate below 50% can still produce a positive average. For a fair coin paid at even odds, the expected gain is zero, and no sequence of past results changes that. This is a statement about arithmetic, not financial advice.

Using a coin flip as a decision tool

Because a fair flip gives equal chances, it works for splitting a tie between two options. A few habits keep it fair:

• Name the sides before the flip, not after.
• Flip once and accept the result, or agree in advance on a best-of-three.
• For more than two options, a coin is the wrong tool. Use a random number generator, or the 1 to 6 version for a single die.
• If you need to pick several distinct items, a generator without repeats prevents duplicates.

If you are choosing between tools for flipping, we compare the options in best coin flip websites and tools.

Run the experiment yourself

We build GizmoBench as a collection of browser-based tools that run locally, with no account or installation. The coin flip simulator lets you flip repeatedly, keep tallies and watch streaks, which suits a classroom demonstration or a sanity check on the numbers above. For the exact math, pair it with the binomial distribution calculator. Other everyday calculators, such as percentages and unit conversions, are collected on the everyday tools page.

A suggested exercise: flip 10 times, record the heads, and repeat ten times. You will see counts between roughly three and seven in most rounds, matching the table above, and probably one round that looks odd. That odd round is the point of the exercise.

Common mistakes to avoid

• Confusing a sequence with a next flip. The chance of a long run is small before it starts. Once it has happened, the next flip is still 1/2.
• Expecting exactly half. Five heads in ten flips is the single likeliest result, at 24.6%, so most ten-flip runs miss it.
• Treating one run as proof of bias. Streaks of four or more appear often in 50 flips, as the 83% figure shows.
• Mixing odds and probability. “1 to 1” and “1/2” describe the same fair flip but are not interchangeable notations.
• Forgetting the model. The formulas assume a fair coin and independent flips. A trick coin or a controlled flip breaks the assumption.

FAQ

Is a coin flip actually 50/50?

In the standard model, yes: heads and tails are each 1/2. Physical coins can deviate slightly. A 2025 study of 350,757 flips, as summarized by MathWorld, estimated that the starting-up side landed up about 50.8% of the time. For everyday decisions, 50/50 is accurate.

What are the chances of losing a coin flip 10 times in a row?

For a specified ten-flip run, it is (1/2)^10 = 1/1,024, about 0.098%. That is rare for a stated run, but over many flips such a run can show up somewhere. The longer the session, the more likely it becomes.

How rare is it to flip tails 7 times in a row?

A specified run of seven tails has a probability of 1/128, about 0.78%. If you flip many times and look for any seven-tail run, the chance is higher, because every flip is a possible starting point.

Does a streak make the opposite result more likely?

No. Each flip is independent, so earlier results do not affect the next one. The ANU explanation says a run of heads does not make tails more likely.

How many heads should I get in 100 flips?

The expected count is 50, with a standard deviation of 5. By our binomial calculation, about 96% of runs land between 40 and 60 heads, so a result like 57 is ordinary.

Sources

• U.S. Mint: Flipping Out for Coins
• CEMC, University of Waterloo: Probabilities of Single Events
• ANU: Heads or Tails? An ANU mathematician breaks down the odds
• Stanford: Probability for Computer Science, many flips
• LSE: Philosophico-Scientific Adventures, lecture 21
• Wolfram MathWorld: Coin Tossing
• The Feynman Lectures on Physics, Vol. I, Ch. 6: Probability
• MIT OpenCourseWare: Introduction to Probability and Statistics
• Trading Expectancy Formula

Recommended

• Coin Flip Simulator: Fair Flips, Tallies and Streaks
• Dice Probability Explorer: Exact Sums & Advantage
• Probability Calculator: Two Events, Venn and Conditionals
• Binomial Distribution Calculator: PMF, CDF and Tails