Probability is the third tool in our kit, and it is gentler than it sounds: a whole pie, sliced so the pieces add to one. The one move that makes quantum mechanics tick is how an amplitude — one of the arrows from the last two pages — turns into a slice. You square it. This page is mostly about why that one move is forced on us, because that "why" is the gap most people fall into.
If something must turn out one way or another, draw a full pie and hand each outcome a slice. A big slice means likely; a sliver means unlikely. The one ironclad rule: the slices must fill the pie exactly — no gaps, no overlap, nothing left over. Add them up and you always land on one whole. A probability is just the fraction of the pie an outcome owns: 0 means never, 1 means always, 0.5 means half the time.
Order one pizza for a table of friends. However you cut it — two huge slices, or twelve slivers — the slices always add back up to one pizza. You can never serve more pizza than you have, and you can't serve a negative slice. Probability obeys the exact same bookkeeping: every outcome gets a slice, the slices are never negative, and together they make exactly one whole.
Before any dials, let us do one by hand at a snail's pace. Suppose a state has amplitudes α = 0.6 and β = 0.8. Here is every step to get the odds:
Now make it move. The single dial θ sets the two amplitudes α and β. Square each and you get the two slices, which always fill the ring. Push the α < 0 button to make an amplitude negative and watch the pie not flinch — squaring erased the sign. This is exactly the machine a qubit runs; you are meeting it one page early.
Here is the question almost everyone skips past. We need a recipe that turns an amplitude into a probability. Whatever it is, it must obey two non-negotiable rules:
"Squaring" passes both. "Just take the size" (drop the sign but don't square) passes rule ① but fails rule ② — and that failure is the whole reason nature squares. The widget below lets you watch it fail. Tilt the state arrow and compare the two recipes side by side.
That locked green 1.00 is no coincidence — it is the Pythagoras rule from page 0.1. For an arrow of length 1, its two shadows obey α² + β² = 1 at every angle. Squaring is the one recipe whose total is welded to the arrow's length, and the length never changes. The "size rule" total, by contrast, swings between 1 and about 1.41 as you tilt — useless as a probability. Nature squares because only squaring keeps the books balanced.
These two are secretly one. Demanding the pie stay full, α² + β² = 1, is the very same thing as demanding the amplitude arrow have length one. "Normalize the state" and "make the odds add up" are not two jobs — they are a single requirement wearing two hats. (The little bars |x| mean "size of x"; for plain real numbers that is just "drop the minus sign," and squaring drops it anyway.)
"The probability is the amplitude." No — you must square it. An amplitude of 0.8 is not an 80% chance; it is a 64% chance, because 0.8² = 0.64. The amplitude and its probability are different numbers.
"Then probabilities can be negative, since amplitudes are." Never. The minus sign lives on the amplitude, and the very act of squaring destroys it: (−0.6)² = 0.36, a perfectly positive slice.
"So the sign is pointless." Far from it. The sign is invisible to a measurement, but decisive before one — it is exactly where two amplitudes can cancel out. Hold onto that tension; it is the entire difference between the quantum world and a roll of dice, and the next page makes it move.