The Math Beneath · Part 2 / 8

Probability is a squashed ruler

Equal steps in probability are not equal amounts of belief. Two moves — odds, then log — straighten the ruler, and the sigmoid appears as nothing more than the trip back.

lives between 0 and 1. That sounds tidy — until you notice that equal steps on this ruler don’t mean equal amounts of anything.

Going from 50% → 59% is a shrug. Still basically a coin flip.

Going from 90% → 99% is enormous — your failure rate just dropped ten-fold, from 1-in-10 to 1-in-100.

Same nine points of distance. Wildly different real change. All the “very sure” territory is crushed into a sliver near 1, and all the “very unlikely” territory into a sliver near 0. Probability is a squashed ruler — the wrong scale for measuring strength of belief. We want a ruler where equal steps mean equal strength. It takes exactly two moves.

Move 1 — odds: remove the ceiling

Stop counting “9 out of 10.” Count wins per loss: 90% becomes 9-to-1, 99% becomes 99-to-1. Now the 90→99 jump shows its true size — the grew more than ten-fold. The scale runs 0 to infinity; nothing is crushed against a ceiling anymore.

But it’s lopsided. “Likely” gets the whole range from 1 to ∞, while all of “unlikely” is squished between 0 and 1. And 4:1 versus 1:4 — which should be perfect mirror images of each other — read as 4 and 0.25. Not mirrors.

Move 2 — log: remove the lopsidedness

Chapter one’s move: when a scale multiplies, take the log. is just log(odds), and everything snaps into place:

4:1 → +1.4 and 1:4 → −1.4. Mirrors at last.
50/50 → odds of 1 → exactly 0. Perfectly unsure sits at zero; belief above, disbelief below.
Doubling your odds is the same-size step everywhere — from 1:1 to 2:1, or from 8:1 to 16:1.

Symmetric around zero, unbounded both ways, additive. This is the straightened belief ruler.

Try it: press double the odds over and over. On the log-odds ruler you take identical strides; on the probability ruler your steps shrivel as you approach 1 — the same belief-growth, squashed. Then hit flip belief from 90%: log-odds mirrors cleanly around 0 while odds lurches from 9:1 to 1:9.

The sigmoid is the trip back

If log-odds is the straight ruler, one question remains: given a spot on the straight ruler, where was I on the squashed one? That return trip has a name — the .

Go deeper: deriving the sigmoid in three lines

Say a score z is the log-odds of some event: log( P / (1−P) ) = z. Undo the log, then solve for P:

P / (1−P) = e^z ⟹ P = e^z / (1 + e^z) = 1 / (1 + e^−z)

That’s the sigmoid — not a curve someone liked, but the only shape consistent with “this score is a log-odds.” Its S-shape is exactly the squash: huge strides far from zero barely move P, because out there the probability ruler is crushed.

You met this exact object in the preferences chapter: the model needs score differences — unbounded, sign-flippable, same-meaning-everywhere — to drive a probability. Those are precisely the three properties of the log-odds ruler. The sigmoid was never picked. It was forced.

Next: what makes beliefs move along this ruler — evidence.

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