If that’s not the answer you expected, you’re not alone. Evidently, the Dilemma, which we’ll call Monty, fools almost everyone at first. This piece steers the problem in a democratic direction, but many explanations of the math have been written and many subtleties explored. For the curious, I recommend:

We’ll go through a series of Monty Hall Dilemmas that parallel the information issues confronting our democracies today. The problem is inherently slippery, and missing or changed context transforms it completely. Explode it out into a group decision, and Monty sheds light on the difficulties of forging individual perspectives into smart collective choices. If combining our knowledge doesn’t elevate it, can democracy solve our trickiest problems? I think it can, but only with a higher bandwidth for processing information.

Monty the Illusionist

The Dilemma is a probabilistic illusion akin to the optical ones we grew up with and are still discovering. People get it wrong unreasonably often. Marilyn vos Savant’s original column apparently prompted “thousands” of letters insisting she’d made a mistake. Responders remained adamant despite additional explanation columns by vos Savant. Formal studies show a strong cross-cultural tendency to stick with the original door on the first try.¹

Our tendency to get this wrong is so powerful that “The Monty Hall Illusion” is an appropriate name. Some part of the problem abuses our probability instincts like a virtual reality headset fools our sense of space.

Monty gives us everything we need to know — we ought to apply a few grade school probability lessons and be done with it, but that’s not what happens. So, Monty’s first lesson: sometimes we’re bad at solving problems despite having all the right information and skills.

The Half Monty

Even with complete context, Monty is deceptive. With puzzle pieces missing, it’s impossible. Imagine your friend walks in half way through the game: she sees two closed doors and one door opened to a goat. Monty explains that one of the doors hides a car, that you’ve chosen one of the closed doors, and that you now have the opportunity to switch. He doesn’t tell your friend when you chose your door or when he opened the goat door. Now he asks her to choose, should you switch your choice?

Without knowing if you chose before or after the goat door was opened, she can’t solve the problem. If she believes you chose after the goat door was opened, the odds really are fifty fifty from her perspective. Monty reminds us of a second lesson: lacking even a small amount of context can put a problem’s solution out of reach.

The Full Monty

Now replace your friend with the entire audience: the decision of whether or not to switch doors will be put to a vote. Let’s assume the audience can come and go as they please, that the problem is new to them, and that the experiment will be run just one time. In this democratic variation, Monty’s two previous lessons imply that getting a majority “switch doors” vote from the audience is a tall order.

Sometimes we’re bad at solving problems despite having all the right information and skills. Since the audience members don’t all arrive on time, only some of them witness the beginning of the game. Others arrive early but don’t pay attention. Some do listen carefully from the beginning, but as we’ve seen, a likely majority of these still won’t vote to switch.

Lacking even a small amount of context can put a problem’s solution out of reach. Audience members that arrive late or don’t scrutinize the process may lack the information necessary to discover a “switch” vote’s advantage.

For audience members in isolation, then, it’s reasonable to expect less than half will vote to switch doors. But they need not be isolated: audience members can talk to each other freely, yell things out, maybe even throw together some signage. If some audience members know switching is best, could the minority convince a majority to vote “switch”? With an easier problem, maybe they could, but Monty’s tricky. In her original columns, Marilyn vos Savant’s thoughtful explanations failed to convince much of her audience. Her counterparts in our experiment would have the same challenge but none of vos Savant’s intellectual reputation or public platform. They might persuade some to switch to switching, but unless they convince a majority, the whole effort is moot.

I don’t have an empirical study to cite, but given Monty’s uncanny difficulty, I doubt that a voting audience can solve the riddle.