Is a Coin Flip Really 50/50? What Real Flips Found

The short answer

• Real coin tosses are very slightly biased, toward the side that starts up. In an experiment of 350,757 flips by 48 people, coins landed on the side they started on 50.8% of the time. Heads and tails came up almost exactly equally often. That matches a 2007 physics model: its often-quoted 51% is its average estimate, 50.8%, rounded up.
• You will not notice it by flipping. In 100 flips, a standard test flags a coin with a 50.8% same-side chance only 6.15% of the time, barely more than the 4.43% at which it flags a perfectly fair coin. To catch the bias 80% of the time takes about 24,000 flips.
• It only helps someone who knows the starting side. If the caller sees which side is up, calls it, and the coin is caught in the palm without being turned over, the caller wins 50.8% of single tosses: roughly one extra win in 125 tosses (51.75% of best-of-7 series). Slap the coin onto the back of the other hand and the small advantage goes to the other side. If the starting side is hidden and as likely to be heads as tails, the caller wins exactly 50%, whatever the bias.
• The virtual coin has no starting side. The yes or no wheel and the Heads or tails preset on the who goes first page pick each side with probability exactly 1/2, independently of every earlier spin.

The figures come from exact binomial math and from 305.3 million simulated coin flips through random.js, the file that picks every result on Wheel Cue; compared at 26 points, all 26 agreed within four standard errors. Percentages are exact unless the sentence says "simulated"; the papers' own figures are credited to them.

What 350,757 real flips found

The idea starts with physics. In "Dynamical Bias in the Coin Toss" (SIAM Review, vol. 49, no. 2, 2007; DOI), Persi Diaconis, Susan Holmes and Richard Montgomery showed that a flipped coin does not just tumble: it also wobbles (precesses), so it spends slightly more of its flight with the starting side up. For natural flips caught in the hand, they put the chance of coming up as started at about .51; the paper explains that this is the average of the estimates from 27 of their filmed flips, 0.508, rounded up. They also pointed out why nobody had checked: estimating a chance near 1/2 with a standard error of 0.001 takes 250,000 flips. (A standard error, not a margin: at that size the 95% margin is still about ±0.002.)

The test came in "Fair Coins Tend to Land on the Same Side They Started: Evidence from 350,757 Flips" by František Bartoš, Eric-Jan Wagenmakers and 48 co-authors, published in the Journal of the American Statistical Association (vol. 120, no. 552, pp. 2118–2127, 2025) after a preprint first posted in 2023. In all, 48 people flipped coins of 44 different currencies and denominations. Each flip started from the side the previous one landed on, every coin was caught in the hand, and the sessions were filmed.

• Same side: 178,079 of 350,757 flips, a share of 0.5077 (95% interval 0.5060 to 0.5094). The paper's headline figure, 0.508, is this share rounded to three decimals. On our arithmetic from the paper's counts, that is about 2,700 more same-side landings than an exact half, 9.12 standard errors above it.
• Heads: 175,421 flips, 0.5001 (95% interval 0.4985 to 0.5018), only 0.14 standard errors from half. The study found no detectable preference for heads or tails.

Two cautions shape everything below. First, 0.508 is the share pooled over all flips, not a fixed property of coins. People differed: in the paper's table of flippers, same-side shares ran from 0.487 to 0.601. The paper's hierarchical model, which lets each person have their own bias, puts a typical flipper at 0.5098 (95% interval 0.5050 to 0.5147), and without the four flippers above 53% the pooled share drops to 0.5060 (95% interval 0.5043 to 0.5077). Second, the bias tended to shrink as people flipped more: the authors suggest that around 10,000 flips (about 10 hours) might virtually eliminate it, while noting that fatigue or waning attention could also explain the decline.

So this article works with a coin, or a flipper, that lands on its starting side exactly 50.8% of the time, every flip, independently. Real flippers vary, so the calculator further down lets you pick other values. As a check on the simulation, we re-ran an experiment of the study's size 100 times with such a coin: the simulated same-side share averaged 0.5079, and its spread from run to run (simulated SD 0.00088) was in line with the exact 0.00084. With a coin like that, an experiment of the study's size would fail the standard 5% test described below about once in 405 trillion tries.

Who the bias favors: calls, series and the starting side

The bias helps whoever knows the starting side when the call is made. For a coin caught in the palm and not turned over, one toss works out like this:

• The caller sees the starting side and calls it: the caller wins 50.80%, an advantage of 0.80 points.
• The starting side is random and hidden from the caller: the caller wins exactly 50%. The coin is as likely to start heads up as tails up, so whatever the caller says, it then matches the starting side half the time and the bias cancels out. Simulated: 50.03% of 2,000,000 tosses.
• The call is made first, and the flipper then starts the coin with the called side down: the caller drops to 49.20%, and the advantage goes to the flipper. Simulated: 49.20%.

Hiding alone is not quite enough: a flipper who always starts the coin heads up, out of sight, still hands a sliver of the bias to a caller who usually says heads.

Turning the coin over changes the direction. Diaconis, Holmes and Montgomery note that a coin caught and then slapped onto the back of the other hand is turned over once, which gives a bias toward the side that started down. On our 50.8% model that would be 49.20% for the starting side, a figure we derived, not one the paper gives.

Series change little. If the same person sees the start and calls it on every toss, a longer series lets the advantage add up, slowly:

Over many calls the advantage shows up in the totals. A caller who sees the start of 100 tosses expects 50.8 wins, and wins more than half of them 52.4% of the time, against 46.0% with a fair coin. Over 1,000 tosses it is 68.2% against 48.7%.

For one kickoff, one "who goes first" or one chore, the difference between 50.80% and 50% is far too small to notice. It matters when the same people toss again and again and one of them always sees the coin first.

A fairer toss

What the two papers support for a real toss comes down to who knows the starting side.

• Hide the starting side, and make it random. Bartoš and colleagues conclude that for high-stakes decisions "the starting position of the coin is best concealed". It should also be random: Diaconis, Holmes and Montgomery mention friends who shake the coin between cupped hands first.
• Call while the coin is in the air, not before the flipper sets it up. A flipper who hears the call first can start the coin with that side down.
• If the starting side can't be hidden, agree how the coin will be caught. Catching it in the palm favors the side that started up; slapping it onto the back of the other hand favors the side that started down. Diaconis, Holmes and Montgomery add that mixing the two, sometimes turned over and sometimes not, may add a good deal of randomness. Once the starting side is random and hidden, the catch no longer matters: the caller wins exactly 50%.
• Catch or let it land? The sources do not settle this. Diaconis, Holmes and Montgomery suspect a coin bouncing on the floor may be less fair, since it can spin on its edge, but note that people often feel the opposite and that bouncing needs a different analysis; Bartoš and colleagues studied caught coins only.

Could you spot the bias yourself?

Suppose you flip a coin many times, always starting from the side it last landed on, and count same-side landings. The standard check is a one-sided exact binomial test at the 5% level: call the coin biased if the count reaches a threshold that a fair coin reaches at most 5% of the time. In 100 flips the threshold is 59; in 1,000 flips it is 527.

• 100 flips: a 50.8% coin is expected to give 0.8 extra same-side landings, while the count's natural spread (one standard deviation) is 5.0. The test flags it 6.15% of the time and flags a fair coin 4.43% of the time. Simulated: 6.11% and 4.42%.
• 1,000 flips: 8 extra landings against a spread of 15.8. The test flags the biased coin 12.1% of the time and a fair coin 4.68%.

How many flips would it take? To flag a 50.8% coin 80% of the time with the one-sided test takes about 24,000 flips; depending on the test and the target, roughly 24,000 to 41,000. The paper's aside that 10,000 flips take about 10 hours implies about 24 hours of flipping for the smaller number. That is our arithmetic, not the paper's, and the paper suggests the bias may fade with that much practice.

The broad message: a bias under one point takes tens of thousands of flips just to show, and more to pin down. Diaconis, Holmes and Montgomery's 250,000 was for a standard error of 0.001, and the 2025 study collected 350,757.

The virtual coin: exactly 1/2, no starting side

On Wheel Cue, a two-entry wheel has no physics to lean on. When you spin the yes or no wheel, the page draws one random 32-bit number with the browser's crypto.getRandomValues() and takes the remainder after dividing by 2: an even number picks the first entry (Yes on the yes or no wheel, Heads in the who goes first page's Heads or tails preset), an odd one the second.

To avoid bias, random.js discards draws at or above the largest multiple of the wheel's size that fits in 2^32. For 2 entries that multiple is 2^32 itself (4,294,967,296), so no draw is ever discarded: exactly half of all possible numbers are even, and each side has probability exactly 1/2. Each spin uses a fresh draw, so it cannot depend on the one before.

In 20,000,000 simulated spins of a two-entry wheel, the first entry came up 50.007% of the time, a spin matched the one before 50.012% of the time, the four possible pairs came up as evenly as independent flips should, and the discard step fired 0 times.

Exactly 1/2 does not mean "alternating". In 100 spins, the longest run of one answer averages 6.98, and 54.2% of sessions contain a run of 7 or more. Why random feels rigged measures these streaks in detail and explains why they look suspicious.

A hand-flipped coin restarted from where it landed, as in the study, streaks slightly more: it repeats 50.8% of the time, so its longest run in 100 flips averages 7.11, and 57.3% of sessions reach 7 or more in a row (3.03 points more). In everyday tossing people set the coin up however they like, so this extra streakiness need not appear.

When the wheel is the better coin

Use the yes or no wheel when there is no coin to hand, when nobody wants to argue about how a coin was tossed and you can all watch the screen, or when you want the odds on screen and a list of results:

1. Open the wheel. The Answers box holds Yes and No; replace them with two names, or with Heads and Tails. The who goes first page opens with a Heads or tails preset already loaded (there the button is Spin the wheel and the list is Results).
2. Check the line under the wheel: it reads "2 entries", "each has a 1 in 2 chance (50%)".
3. Press Spin for an answer. The result window shows "Probability 50%" and a spin ID, and How this result was picked shows the draw: rejection limit 4294967296, 0 discarded draws (it cannot be anything else with two entries), and the number mod 2.
4. Keep Remove the winner after each spin off (it is off by default) if you will call again. Copy in the Answers so far panel copies every spin with its probability and spin ID.

If you are not in the same room, Send this wheel sends a link with the two entries in it, and whoever opens it spins on their own screen. Nobody else sees that spin, so the spin ID and the copied list are a note for yourselves, not proof. Agree first who spins and that the first spin counts, or share a screen while they spin.

Keep the wheel, like a coin, for choices where either answer is fine.

How to check these numbers

Everything exact here is binomial arithmetic. Best-of series, call scenarios and streaks are exact fractions with the coin's chance written as 127/250. Test thresholds, flag rates and sample sizes are exact binomial sums, checked against exact fractions at 100 and 1,000 flips. The simulation is separate: each virtual flip is one CueRandom.pickWeighted([1, 1]) call, exactly as the wheel spins, and each hand flip of the 50.8% model is one CueRandom.below(1000) call that lands on the starting side when the result is below 508.

• Checks. A check passes when the simulated and exact values differ by less than four standard errors. All 26 passed; the largest gap was 1.62 standard errors. The pair check uses Good's serial test, which suits overlapping pairs of flips.
• The papers' figures. The study figures above are the papers' own, read in the full text of the preprint (version 4) and in the 2007 paper; the journal version is DOI 10.1080/01621459.2025.2516210. The shares, z-scores and the miss chance for a study of that size are our arithmetic on their published counts.
• Re-run it. The script and raw results are linked at the top of the page. Save the script and random.js in one folder and run node is-a-coin-flip-really-50-50.mjs random.js results.json with Node.js. The simulated values will differ in the last decimals; the exact ones will not.
• The code itself. The fairness report tests random.js at scale, and how randomness works explains the method.