Section 5.4

Conditioning Paradoxes

Puzzles where what is revealed is subtle: boy-girl variants, Bertrand's box, painted-cube faces, Monty Hall and its variants, Russian roulette, the added baby.

Toolkit

The common trap is conditioning on the wrong sample space element: compute $\mathbb{P}(A\mid B)$ by restricting to outcomes actually consistent with $B$ as literally stated, not with a superficially similar event, and remember which outcomes are generated by a process that itself depends on counts or labels.

\mathbb{P}(A\mid B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)},\qquad \mathbb{P}(A\mid B)\propto \mathbb{P}(B\mid A)\mathbb{P}(A).

Key points:

Enumerate the underlying equally-likely atoms (sequences, labelled balls, individual faces, individual door-configurations) before conditioning; symmetry arguments (“this outcome is as likely as that one”) must act on these atoms.
When an informant’s reveal is selective (always avoids certain outcomes, as in Monty Hall or Bertrand’s box), its probability of revealing each remaining possibility is not uniform unless shown to be.
When the observation itself is size- or count-weighted (seeing a face, picking a person from a room), weight each hypothesis by how likely it is to produce that exact observation, not just by its prior count.
For sequential random processes (dice rerolled, chambers spun or not), recompute the conditioning event’s probability directly from the process rather than by analogy.

Problems