Persi Diaconis left home at fourteen to travel with a card magician named Dai Vernon. For the next decade he lived on the road, performing sleight of hand, learning to control what audiences believed was uncontrollable. Then he went back to school at twenty-four, not because the magic stopped working, but because he wanted to know why it worked. He ended up at Stanford, where he became one of the foremost mathematicians studying randomness.

He has spent the last forty years proving, in one context after another, that the things we treat as random mostly aren’t.

I am not a mathematician. I am an eighteen-year-old college student who reads too many papers and finds probability theory unreasonably interesting. But the more I’ve read of Diaconis and the people working in his wake, the more I’ve started to notice something: the gap between what we call random and what actually is random is not small. It is large, it is measurable, and it shows up in places so ordinary that you’d never think to look.

Coin flips. Card shuffles. Playlists. Lottery tickets. The things we use as shorthand for “fair” and “unpredictable” and “beyond human control.” None of them are quite what we think.

The Coin Is Lying to You

A coin flip is the canonical example of a fair random process. Fifty-fifty. Equal odds. The thing you do when you can’t decide where to eat.

It’s not fifty-fifty.

In 2007, Diaconis, Susan Holmes, and Richard Montgomery published a physics model of human coin tossing. A vigorously flipped coin tends to land on the same side it started on. The predicted bias: about 51 percent.

You might think that’s too small to matter. For a single flip, sure. But in 2023, a team led by František Bartoš at the University of Amsterdam decided to test the prediction at scale. They recruited 47 people across six countries, handed them coins, and said flip. 350,757 flips total, including one twelve-hour marathon session.

The result: same-side-up 50.8% of the time. The bias is real. It comes from precession, a slight wobble in the coin’s rotation that causes it to spend a fraction more airtime with its starting face up. Not a lot more. Just enough.

The deeper issue is that the coin flip isn’t random at all. It’s deterministic. From the moment the coin leaves your thumb, its trajectory is governed entirely by classical mechanics: initial position, force, angle, angular momentum. Know all of that precisely and you can predict the outcome every time. Diaconis built a machine that does exactly this. Same flip, same result, 100% of the time.

Randomness, in this context, is a word for the things we don’t bother measuring.

If you bet a dollar on a coin flip a thousand times and knew the starting position each time, you’d come out about $19 ahead on average. Not a fortune. Not nothing either. And definitely not fair.

Seven Shuffles, or the Cliff You Can’t See Coming

How many times do you need to riffle shuffle a standard 52-card deck to make it random?

Seven.

In 1992, Dave Bayer and Diaconis proved that after seven riffle shuffles, every possible arrangement of a 52-card deck is nearly equally likely. Fewer than seven, and it isn’t close. More than seven, and you’re not gaining much.

The remarkable part isn’t the number. It’s the shape of the transition.

You’d expect randomness to accumulate gradually. Each shuffle mixes the deck a little more, smooth and linear. It doesn’t work that way. The deck stays largely ordered through shuffles one through five. Then somewhere between five and seven it falls off a cliff. Mathematicians call this the cutoff phenomenon: the deck goes from mostly ordered to mostly random not in a gentle slope but in a sharp break.

Below the cutoff, the deck retains enough structure that a clever person can exploit it. Diaconis describes a card trick where a spectator shuffles three times, picks a card, and the magician can still find it. Three shuffles preserve that much order. After five shuffles, about 3.5% of the original information remains. After six, less than 1%. Seven, and you’re done.

Think about what this means for every poker game you’ve sat through where someone gave the deck two lazy shuffles before dealing. Those cards were not random. The original order was still in there. You just couldn’t see it.

The guy who shuffled twice and dealt you that full house? He didn’t get lucky. He just didn’t shuffle enough for luck to be involved.

Spotify Tried Randomness. People Hated It.

When Spotify launched its shuffle feature, it used a Fisher-Yates algorithm. Mathematically perfect random ordering. Every song in your playlist had an equal probability of appearing in any position. By any technical standard, it was random.

Users complained it was broken.

True randomness produces clusters. Twenty songs by five artists, and a random shuffle might play three by the same artist back to back. Might play your favorite song twice in three days and then not for two weeks. Not a bug. That is what randomness looks like. But it doesn’t look random to humans.

So in 2014, Spotify made their shuffle less random. They replaced the pure algorithm with one that deliberately spaces out songs by the same artist, avoids clustering, distributes tracks more evenly across the playlist. The result felt more random to listeners. It was, mathematically, less so.

What the company admitted, whether they meant to or not, is that what people want isn’t randomness. It’s the aesthetic of randomness, which turns out to be a different thing entirely.

We use “random” as a synonym for three separate properties: unpredictable, fair, and evenly distributed. True randomness only reliably delivers the first.

The Lottery Ticket That Looks Wrong

In a 6/49 lottery, is 1-2-3-4-5-6 as likely to win as 7-19-23-31-38-42?

Yes. One in 13,983,816. Same as every other combination.

Most people don’t believe this, even when they understand it intellectually. Sequential numbers feel patterned. They feel chosen. Our brains are pattern-detection machines running on hardware that evolved to spot predators in tall grass, not to intuit combinatorics.

1-2-3-4-5-6 has never been drawn in a major lottery. Neither has 7-19-23-31-38-42. You just don’t notice the second one, because it doesn’t look like anything.

There is a practical reason not to play 1-2-3-4-5-6, but it has nothing to do with probability. About 10,000 people play that combination in every draw. If it ever hits, the jackpot splits 10,000 ways. You’d win the lottery and walk away with what amounts to a decent used car.

The math doesn’t care which numbers you pick. The economics care enormously.

Where the Gap Gets Expensive

Most of the time, none of this matters. The coin is close enough to fair. The deck is close enough to shuffled. The playlist is close enough to random. We live in the margin of error and do fine.

But casinos don’t operate in the margin of error. Neither do cryptographers, or the people running clinical trials, or anyone building systems where the difference between perceived randomness and actual randomness is worth something. Diaconis himself has been hired by casino executives to audit their automatic card-shuffling machines. He found flaws. They were not pleased.

The more you think about randomness, the less random things become. Diaconis said that. He meant it about his research. I think it applies more broadly.

Anyway. Next time someone settles a decision with a coin flip, maybe glance at which side is facing up.