How to explain the Monty Hall problem to a disbeliever
The vital part about this problem is that the presenter knows which box is the right one. By opening an empty box, he&#x27;s giving you extra information. And that means there&#x27;s 2&#x2F;3 chance of the remaining box having the prize.<p>If, on the other hand, the presenter didn&#x27;t know which box is the right one, and just opens one of the other two boxes at random (with a 1&#x2F;3 chance of opening the box with the prize), then, if the opened box turns out to be empty, the chance of the remaining box being the right one drops to 50%.<p>The difference between these two scenarios becomes obvious if we expand it to 100 boxes: You pick a box, 1% chance of being right. Of the other 99, at least 98 are empty. The presenter knows which, opens the 98 empty boxes, and now there&#x27;s a 99% chance of the remaining box being the right one.<p>Other scenario: The presenter opens 98 boxes, not knowing which are empty, so he has a 98% chance of opening the box with the prize. On the unlikely chance that all are empty, there prize must have been in either the box you picked, or the remaining box, but we still have no information about which it is, so there&#x27;s a 50% that you&#x27;re holding the right box.<p>Of course if you don&#x27;t know whether the presenter knows, and you don&#x27;t know if he was lucky that the 98 boxes he opened were all empty, or that he knew, then the situation becomes quite a bit more complicated, but the chance of switching being the best option is going to be larger than 50%.<p>How much? Let&#x27;s say there&#x27;s an a priori 50% chance that he knows or doesn&#x27;t know. If he doesn&#x27;t know, then opening 98 empty boxes is pretty unlikely, so it&#x27;s pretty likely that he knows. It&#x27;s probably possible to calculate those odds, but I&#x27;m not going to try that now.<p>And if the presenter&#x27;s behaviour changes depending on whether you picked the right box or not, for example he uses his knowledge to actively tempt you away from the right box, then all bets are off. Or maybe him opening another box is proof that you&#x27;ve got the right one. Or maybe that&#x27;s what he wants you to think...