Why the unopened doors are not equally likely
The puzzle feels as if it resets after a losing portal is revealed: two portals remain, so surely each has a 50 percent chance. But the reveal is not a random event. The host is choosing what to show while avoiding the prize, which means the host’s action depends on where the prize is.
Imagine sorting the game by your first choice. In one-third of games, you picked the prize. The host must then open one of the losing portals, and switching takes you away from the prize. In the other two-thirds, your first pick was wrong. The host has no choice but to open the other losing portal, leaving the prize behind the only alternative you can switch to. Switching therefore wins in those two cases out of three.
The reveal carries information
A losing portal being opened does not change where the prize was. It changes what you know about the alternatives. The host’s rule filters the possibilities: a portal that could have been opened by chance is now known to have been selected specifically because it was safe to show.
The same logic becomes easier to see with 100 portals. You choose one; the host, who knows where the prize is, opens 98 losing portals and leaves your original choice and one other. Your first pick was unlikely to be right. The host’s careful removal of losing options makes the remaining alternative look compelling—not because it gained probability on its own, but because the host’s action points toward it.
The rules are part of the answer
Switching has this advantage only under the stated setup: the host knows where the prize is, always opens a losing portal, and always offers the chance to switch. If the host opens portals randomly, might reveal the prize, or offers a switch selectively, the reveal means something different. The furious part of the puzzle is that the arithmetic is simple; it is the host’s hidden constraints that people tend to overlook.
