Inverse Probability: You Can't Be Indifferent About It!

Core idea

A conditional probability and its inverse point in opposite directions. P(o|g) — the probability of order given a creator — is not the same as P(g|o) — the probability of a creator given order. Confusing them is a classic mistake: P(Australia | wild kangaroo) is very high, while P(wild kangaroo | Australia) is low.

The two are linked by a fixed equation. Because conditional probability is defined as P(a|b) equals pr(a and b) over pr(b), and a and b says the same thing as b and a, a little algebra yields Bayes's theorem: P(g|o) equals P(o|g) times pr(g), all divided by pr(o). The theorem tells you how to update a prior probability into a posterior once evidence arrives.

Priest's argument: An inductive argument from evidence to hypothesis succeeds only if the hypothesis's prior probability is high enough — and the Argument to Design quietly assumes a prior it has no right to.

Why it matters

Why the Argument to Design fails

The Argument to Design says the ordered cosmos o is good reason to believe in a creator g. It is true that P(o|g) far exceeds P(o| not g) — order is likely if there is a designer. But what the argument needs is P(g|o) greater than P(not g|o). Run that through Bayes's theorem and the conclusion survives only if the prior pr(g) is at least as great as pr(not g). There is no reason to think so. There are vastly many ways the cosmos could have been, and very few are significantly ordered — that is exactly what gives the argument its bite. So a priori a creator is the less likely hypothesis. The argument seduces only by confusing a probability with its inverse.

When inverse reasoning works

Inverse probability is not doomed — most inductive arguments use it well. Two roulette wheels, one secretly biased toward red; you watch wheel A spin and update on the result sequence. Here the priors are genuinely settled: with no reason to favour A over B, each gets prior 1/2. Bayes's theorem then lets the observed spins tip the balance toward "A is fixed". The difference from the Design argument is entirely in whether the priors can be defended.

Key takeaways

Mental model

Copy sourceZoom

Practical application

When evidence seems to support a hypothesis, do not stop at "this evidence is likely if the hypothesis is true". Ask what the hypothesis's prior probability was before the evidence.

Example

A car leaves at noon for a town 300 km away, travelling at a constant speed between 50 and 100 km/h. By the Principle of Indifference, the midpoint time is 4:30 p.m., so the car is equally likely to arrive before or after 4:30. But the midpoint speed is 75 km/h — equally likely above or below — and 75 km/h gives an arrival of 4:00 p.m., so it is equally likely to arrive before or after 4:00. The two applications of one principle contradict each other: the car cannot be equally split at both 4:00 and 4:30. The Principle gave a different answer depending on whether we carved the problem by speed or by time. That is the topic's closing crack: indifference, indispensable to everyday probabilistic reasoning, has no built-in rule for which parameter to be indifferent about.

Related lessons