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One dot splitting into two, then four, then eight endings A single filled dot on the left grows two straight branches to two dots. Each of those grows two more branches to four dots, and each of those grows two more branches to a column of eight short blocks down the right-hand side. The number of endings doubles at every step from one to eight.

How to Count What You Cannot List

About 11 minutes

Five heads in a row. Is that remarkable, or is it Tuesday?

The tool for answering it is already in your hands, because it is the same tool that made seven a hill: count how many ways things can come out, then count how many of those ways are the one you care about. What is new is that you can no longer draw the ways. There is no grid with a square for every possible run of five coins that fits on a page — and there does not need to be.

Every new coin doubles everything

Start small enough to be certain.

One coin has two results: heads, tails.

Two coins. Take each of those two and add the second coin's two results to it. Heads becomes heads-heads and heads-tails. Tails becomes tails-heads and tails-tails. Two results turned into four.

Three coins. Take each of the four and split it in two the same way. Eight.

That is the whole engine, and it never changes. Adding a coin does not add a few more results. It takes every result you already had and splits each one into two, so the number doubles.

A coin-flip tree doubling from two endings to eight A single dot at the top centre grows two straight branches down to two dots, one for heads and one for tails. Each of those grows two more branches to four dots, and each of those grows two more to eight dots along the bottom. Down the left-hand side a number gives the count of endings on each row: two, then four, then eight. Every row has twice as many endings as the row above it. 1 coin22 coins43 coins8 start every row has twice as many endings as the one above it
One coin, then two, then three. Each dot on a row grows two branches, so each row has twice as many endings as the row above it. Nothing here counts upward — the doubling does all the work.
Coins Ways it can come out
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1,024

Ten coins have 1,024 possible results. Nobody wrote them out. The doubling counted them.

Now answer the question

Five coins have 32 ways to land. How many of those are five heads?

One. There is nothing to work out — heads-heads-heads-heads-heads is a single ending on that tree, and no other ending looks like it.

So five heads in a row is 1 in 32. Not one in a hundred, not one in a thousand. About three flips-of-five in every hundred.

Five tails in a row is another 1 in 32. And "all five the same, either way" is 2 in 32, or 1 in 16.

Five coins, thirty-two ways to land. Drag these into order, from the fewest ways to the most.

  • At least two heads
  • All five heads
  • At least one head
  • Exactly two heads
  • Exactly one head

All five heads is 1 way. Exactly one head is 5, because the head can be any one of the five coins. Exactly two heads is 10 — count the pairs of coins the two heads could sit on. At least two heads is everything except the one all-tails ending and the five one-head endings, so 32 − 6 = 26. At least one head is everything except all-tails: 31.

Two of those are worth staring at. "Exactly one head" and "all five heads" both sound like one specific thing, and one of them is five times more likely than the other, because there are five different coins that could be the head.

The number of tries changes everything

Here is where a one-in-thirty-two stops feeling rare.

Give every child in a class of 30 five coins and have them flip. Each child has a 1 in 32 chance of all heads. Thirty children is very nearly thirty-two tries, so you would expect about one all-heads in the room, and there is a 61% chance at least one turns up.

The child holding it has done nothing unusual. They have done the thing you should expect somebody to do.

Now scale it. Ten heads in a row is 1 in 1,024, which is genuinely uncommon — one class trying once a day would wait about five years. But a school of 500 children, flipping ten coins once a day for a 190-day school year, is 95,000 tries. Expect about 93 perfect ten-head runs.

Nothing has become more likely. There are simply more chances for it to happen to somebody.

You flip four coins. Which single result is most likely?

That quiz catches nearly everyone, and it is worth understanding why. HHHH feels special and HTHT feels ordinary — but as single results they are two endings on the same tree, and every ending on the tree has exactly the same chance. What is uncommon is not HHHH itself. It is the description "all four the same", which only two endings fit, while a description like "two heads and two tails" fits six.

The trap this leaves open

The two things this chapter did were: count the ways, and then count the tries. Both were easy to do here, because both were written on the page.

They are not always written on the page, and the number of tries is the one that hides. Thirty children flipping coins is obviously thirty tries. But some situations contain far more tries than there are people in them, and if you count the people you will get a badly wrong answer.

Twenty-three children stand in a room. What are the chances that two of them share a birthday?