🎲 The chance puzzle
How Likely, Really?
You can give every “maybe” in the world a number, even when your gut feeling says something different.
Your gut has strong feelings about chance.
Some of those feelings are famously wrong.
Let’s find out which ones.
Getting your bearings
Putting chances on a line
You can put any chance somewhere on a single line. 0 at one end means “no chance ever”, 1 at the other end means “dead cert”, and a coin flip goes right in the middle at ½. Everything else goes somewhere in between.
Your turn. Below are six everyday “maybes”. Drag each card to where you reckon it belongs on the line (close is good enough). When you place a card, you’ll see its real chance as a fraction, a decimal and a percentage. (If you tap a card without dragging, it’ll just show you the answer.)
Place all six cards on the line. Each one will show its chance as a fraction, a decimal and a percentage.
A coin has 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. Here are the five flips:
What’s the chance that the next flip is heads? Answer before you flip, and stick with your answer.
Pick one to unlock the coin.
Flip a coin once and it lands heads. Would anyone say the next flip now “owes” you tails? Of course not, because one head doesn’t change anything.
Five heads is the same thing five times over. The coin can’t remember what it did before, so every flip starts again at ½.
You can’t use ½ to predict the next flip.
It tells you what happens in the long run.
The why
What happens in the long run
The brass line shows the share of your flips that have come up heads so far, and the teal line marks ½. Before you press anything, have a guess: after 10 flips, will the brass line be near ½, or a long way off?
Early on, the brass line swings about a lot.
Look out for two things. First, by a thousand flips the line stays very close to ½. That happens because each new flip makes less and less difference to the average, even though the coin never “corrects itself”. Second, look at the longest streak. Runs of five or six heads in a row often turn up in a hundred flips, and that’s normal for random flips.
You can’t predict a single flip.
But what about ten thousand of them?
You can predict those almost to the decimal, because random results follow a pattern.
Doing the sums
Which total comes up most with two dice?
Roll two dice and add them, and you’ll get a total from 2 to 12. The totals are not equally likely, and you can work out exactly why. First, guess which total comes up most often, then roll until the shape of the bars shows you.
Pick your guess to unlock the dice. The teal dashes show the shape the maths expects. Roll and see how close the brass bars get to it.
Each total can come up in a number of different ways, or routes. Here are all 36 ways two dice can land, with one die across the top and 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. If the left one rolls 3 and the right one rolls 4, that’s one way they can land. Left 4 and right 3 is a different way, even though both add to 7.
The grid counts the different ways the dice can land. That’s why it has 36 squares, and 7 comes up most because it has six of them.
The forecast says there’s a seventy per cent chance of rain, and the day stays dry.
That doesn’t make the forecast wrong, because on three days in ten like that, it stays dry.
A forecast can be right even when the day doesn’t go the way it said.
One more thing
Now you’ll start spotting chance everywhere. The forecast can give an 80% chance of rain and the day can still stay dry. If you’ve ever lost a board game to somebody’s jammy run of sixes, that was just chance. Runs like that happen more often than people expect.
And next time someone says tails is due, ask them: how would the coin know? If you can explain why it can’t, you really understand chance.
The garden where numbers grow · chance, from 0 to 1