Subject: Re: Multi-stage Monty Hall problem
Date: Sun, 14 Dec 1997 23:00:05 -0500
From: Alex Bogomolny

SWITCH-STICK:

To win, the fellow must first miss (.75) and then guess right (.5) when switching. .75*.5 = .375

SWITCH-SWITCH

There are two independent possibilities:

  1. The fellow picks right (.25)
  2. The fellow misses on first pick (.75)

Looking into each case separately:

  1. After he picked right, Monty Hall opens an empty door, the fellow abandons the prize and switches to an empty door. Monty Hall opens the last empty door. Therefore to switch means to open the original door that was the right guess to start with. The whole sequence thus has the probability of .25.
  2. After he picked wrong (.75), he may pick wrong again (.5) so that the second switch necessarily leads to a win. This sequence has the probability of .75*.5 = .375.

Summing up .375 + .25 gives .625.

You may want to try similar analysis with five doors. Then you'll be able to generalize the best strategy: STICK until the very last and then SWITCH. Also note that the more doors there are the better this strategy works.

Best regards,
Alexander Bogomolny

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