Why Fewer Assumptions Win

Imagine you’re building a tower out of glasses, stacking them one on top of another. One glass? Rock solid. Add a second, balanced on the rim of the first — a little wobbly. A third, a fourth, a fifth — and now you’re holding your breath, because the whole tower only stands if every single glass stays put. Knock one over and the entire thing comes down.

That’s an explanation with lots of assumptions. Every assumption is another glass. The story only “stands” if all of them are true at the same time — and as you’ll see in a moment, the more you stack, the lower the odds the whole thing holds.

In the last lesson you met Occam’s razor: when two explanations fit the facts equally well, prefer the one that needs fewer assumptions (an assumption is something an explanation quietly requires to be true — a claim you’re taking on faith for the story to work). That lesson told you what the razor says. This one shows you the engine underneath it — and the engine is just multiplication.

The conjunction rule: “and” is expensive

Here’s the key move. To believe a whole explanation, you have to believe all of its assumptions are true simultaneously. Not one of them. Not most of them. All of them, at once.

That word and is doing heavy lifting. “The package is late and the courier lost it and the warehouse mislabeled it” is a single story made of three separate claims, and it’s only correct if claim 1 and claim 2 and claim 3 all hold. In probability, that “all-at-once” combination is called a conjunction — two or more claims joined by “and.”

For claims that don’t affect each other’s odds (we call those independent — knowing one tells you nothing about the other), the probability of the conjunction is the claims multiplied together:

$P(A \land B \land C) = P(A) \times P(B) \times P(C)$

(The little $\land$ symbol just means “and.”)

Now here’s the part that makes the whole razor work. A probability is always a number between 0 and 1 (0% to 100%). And when you multiply numbers that are less than 1, the answer only ever gets smaller.

The worked example: watch a plausible story sink below 50%

Let’s make this concrete with real numbers. Suppose something happened and you have two competing explanations.

• Lean explanation: needs just one assumption to be true, and that assumption is 90% likely ($0.9$).
• Heavy explanation: needs three assumptions, and — here’s the trap — each one sounds quite reasonable on its own, 70% likely ($0.7$).

Seventy percent feels safe, right? You’d take a 70% bet. But the heavy story needs all three 70% claims to land together. Watch what the conjunction rule does to it:

Look at that. Each individual 70% assumption sounded perfectly believable, but stacking three of them drags the whole story down to 34% — below even odds. The lean explanation, at 90%, isn’t just a little better; it’s nearly three times as likely to be the true account.

And it gets worse the more you stack. Watch a story made of four assumptions that are each very believable — 80% likely:

Four claims, each 80% solid, and the complete story is still only a coin-flip-ish 41%. That’s the razor in action: the elaborate explanation didn’t lose because any one piece was weak. It lost because believing all the pieces at once is hard, and the math charges you for every “and.”

Fewer assumptions = fewer ways to be wrong

There’s a second way to see the same truth, no math required. Think of each assumption as a failure point — a place the story could snap.

A one-assumption explanation has exactly one thing that has to go right. A five-assumption explanation has five separate things that all have to go right, which means it has five separate ways to be wrong. Add a link to a chain and you’ve added a place it can break.

This is why simpler explanations aren’t just more elegant — they’re more robust. There’s simply less of them that can be mistaken. (Engineers think this way constantly: a design with fewer parts has fewer parts to fail. Some people call the deliberate cousin of this idea a margin of safety — leaving room for the inevitable broken link. There’s a whole course on that idea down the road; for now, just notice the shape: more parts, more fragility.)

The conjunction fallacy: why detail seduces us

Now flip the camera around. The math says adding claims can only lower a story’s probability. But our intuition does the exact opposite — it finds detailed, vivid stories more convincing. That backwards instinct has a name.

In a famous experiment, psychologists Amos Tversky and Daniel Kahneman described a woman:

Linda is 31, single, outspoken, and very bright. As a student she was deeply concerned with discrimination and social justice, and she joined anti-nuclear demonstrations.

Then they asked which is more probable:

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

Most people picked 2. It just feels righter — the activist detail fits the story of Linda so well. But pick 2 and you’ve made a mathematical impossibility. Option 2 is a subset of option 1: every Linda who is a feminist bank teller is, by definition, also a bank teller. The “and” can only narrow the group, never widen it. So option 2 can be at most as probable as option 1 — and almost always less.

This mistake — judging a specific, detailed conjunction as more likely than the broad claim it lives inside — is the conjunction fallacy. It’s our brain getting fooled by fit: the added detail “feminist” matches the vivid description, so the combo feels more plausible, even though every detail you bolt on multiplies the odds down, exactly as the table above showed.

How this connects to the rest of your toolkit

This lesson isn’t a standalone trick — it’s the hinge that several ideas turn on:

• Thinking in probabilities / base rates. This is probability applied to explanations. Instead of asking “could this story be true?” you ask “what are the odds the whole story is true at once?” — and you answer it by multiplying. A story’s plausibility is a number, not a vibe.
• Occam’s razor (last lesson). Now you know why the razor cuts. “Prefer fewer assumptions” isn’t a stylistic preference; it’s a direct consequence of the conjunction rule. Fewer assumptions literally means a higher joint probability, all else equal.
• Confirmation bias. A detailed story that agrees with what you already believe feels even likelier — that’s the conjunction fallacy weaponized by your prior opinions. The richest, most satisfying conspiracy is usually the one that flatters your existing worldview, and every “and” in it is quietly draining its real probability.

The honest fine print

Two caveats, because a good mental model knows its own edges.

First: assumptions aren’t always independent. The clean multiplication rule — $P(A \land B) = P(A) \times P(B)$ — only holds exactly when the claims don’t influence each other. In the real world they often do: if “the flight was canceled” is true, then “I missed my connection” becomes much more likely, so you can’t just multiply their standalone odds. When claims are correlated, the precise number will be off.

But — and this is the important part — the qualitative conclusion survives intact. Even with correlated assumptions, piling on more required claims can only ever hold the joint probability flat or push it down. It can never push it up. So treat the multiplication as a model: the exact figure is an approximation, but the lesson (“more assumptions → not more likely”) is rock solid.

Second: a lower-probability explanation can still be the true one. This is the course’s recurring drumbeat. Probability is a prior, not a verdict. The simplest explanation is the smart default bet, the place to start — not a guarantee. Sometimes the elaborate, unlikely story is exactly what happened; rare things do occur, and the universe is under no obligation to be tidy. The razor tells you where to put your chips, not how the hand will play out. Hold your simple explanations firmly, but not so tightly that real evidence can’t pry them loose.

Recap

You now know the gears turning inside Occam’s razor: every “and” is a tax, and the bill is paid in probability. Next up — Hanlon’s razor, the razor’s mischievous sibling, which asks why we so often reach for an elaborate story of malice when a boring one about a simple mistake would do.