The reason as most people do not see is that the game changes once you block out an unknown door, which has only a 1/3 chance of being a cadillac and THEN a door which has 0 chance of being a Cadillac after it is opened for you. That means that the chance if 100% that the Cadillac is behind the door you selected -AND- the door that was not opened. It is now 33.33 per cent chance that the Caddie it behind your door as it always was, and 66.67% chance NOW THAT YOU kNOW WHERE ONE OF THE GOATS IS, that is NOT behind the door you selected. It ALWAYS is that SAME chance -- that does NOT CHANGE ... but now you know WHICH DOOR to change TO!!!!!!!!!!
So we know that the chance it is behind the two doors, yours and the unselected door is 100%. Which is 33% for your "picked" door, and what must it be for the unselected door? 33.33+66.67 =100% So it is 66.67% chance to be behind the unselected door.
Think of it this way. What chance is there that given any door, you have a goat behind it? 66.67% right? So when a door is opened and it shows a goat, the chances for the remaining doors have changed, right? The doors remaining must be Cadillac-goat, but they can be g-c or c-g. But remember the chance you landed on a goat is NOT 50%! It is 66.67% right?! So the chance you see are not g-c, c-g but G66-C33, C33-G66. The remaining doors, i.e. the one you selected and the one that was not opened that you did not select, can only be two ways, g-c or reverse, but which one did you land on most often, the g or the c? It is NOT even!!!!
No matter where the Caddie is of the other two doors that are not opened, you landed on the goat most often. It sounds crazy, but what it means is that it is twice as likely that you landed on a goat as a Cadillac from the get-go, so the distribution you imagine for the remaining two doors is never 50-50. It nore often puts you on the goat, it is never even.
The key is the first selection. It is 66% goat so to speak. Then when you add a goat to that duple you have one goat + a 66% chance for a second goat. So 66% chance you have two goats in that "fixed" duple. A goat plus one other door, IF the unknown door is first selected but not opened.. order matters... IS 66% goat-goat..
So it follows that it is 33% chance the remaining door is NOT a goat door.
If I select a known goat-door, and then bid you pick, it goes back to 50-50. You select first, it is 33-66.
EC<:-} |
