🎲 The chance puzzle
How Likely, Really?
You can pin every “maybe” in the world to a number. Your gut disagrees. Loudly.
Your gut has strong opinions about chance.
Some of them are wrong. Famously wrong.
Time to catch it in the act.
Getting your bearings
Every maybe has an address
Here’s the whole trick. Any chance lives somewhere on a single line: 0 at one end for “no chance ever”, 1 at the other for “dead cert”, and a coin flip sitting bang in the middle at ½. Everything else is a spot in between.
Your turn. Below are six everyday “maybes”. Drag each card onto the line to where you reckon it belongs (close counts). Every card you place tells you its real number in all three costumes: fraction, decimal, percentage. (Tap a card without dragging and it tells you straight.)
Six cards, one line. Each one you place shows its number in all three costumes: fraction, decimal, percentage. Same number, three outfits. You know that trick already.
A coin just landed heads five times in a row.
Surely tails is owed now?
The trap
The coin with no memory
This coin has just landed heads five times on the trot. Genuinely. Here it is, mid-streak:
So: the chance the next flip is heads. Answer before you flip. No take-backs.
Pick one to unlock the coin.
Flip a coin once. It lands heads. Would anyone say the next flip now “owes” you tails? Course not. One head means nothing.
Five heads is that same nothing, five times over. The coin keeps no diary, holds no grudges, and owes you nothing. Every flip is flip number one.
½ was never a promise about the next flip.
It’s a promise about the long run.
The why
The long run keeps the promise
Watch the coin keep the promise. The brass line is the running share of heads in your flips; the teal line is the promise of ½. Before you press anything: after 10 flips, do you reckon the brass line sits near ½, or somewhere wild?
Early on the brass line swings about wildly. That’s the point.
Two things to catch. One: by a thousand flips the line hugs ½. Not because the coin “corrects itself”, but because each new flip matters less and less to the average. Two: look at the longest streak. Runs of five or six heads turn up in a hundred flips all by themselves. Streaks aren’t the coin misbehaving. They’re what ordinary randomness looks like.
You can’t call a single flip.
But ten thousand of them?
Predictable almost to the decimal. Randomness has a shape.
Doing the sums
Two dice, eleven suspects
Roll two dice and add them: the totals run 2 to 12. They are not equally likely, and you can work out exactly why. First, pick your suspect (which total turns up most often?), then roll until the shape gives it away.
Pick a suspect to unlock the dice. The teal dashes already show the shape the maths expects. Roll and watch the brass bars chase it.
Every total is a set of routes. Here are all 36 ways two dice can land: one die across the top, the other down the side. Tap any square to light up every route to that total.
Put one die in your left hand and one in your right. Left rolls 3, right rolls 4. That’s one way the world can land. Left 4, right 3 is a different one, even though both add to 7.
The grid counts ways-the-world-can-land, not sums. That’s why it has 36 squares, and why 7, holding six of them, keeps winning.
Seventy per cent chance of rain, and the day stays dry.
The forecast wasn’t wrong. Three days in ten look exactly like that.
A good call and a good result are not the same thing.
One more thing
Chance is loose in the wild, and now you can spot it. A keeper can read a penalty perfectly and still let it in. Every board game you’ve ever lost to somebody’s jammy run of sixes: that wasn’t the dice picking sides. Streaks are randomness doing its ordinary thing.
And next time someone swears tails is due, ask them one question: how does the coin know? If you can explain why it can’t, this one’s yours for good.
The garden where numbers grow · chance, from 0 to 1