A different perspective on the usual puzzle.
Our host M, offers contestant A a choice of three doors - only one of which reveals a winning outcome. A chooses door #1, and M opens door #3, revealing a non winning outcome.
At this point, contestant B, who has been watching from the audience, is invited on stage, and given a choice between doors #1 and #2.
Following this selection, we invite contestant C to the stage. Contestant C knows the rules of the game, but was absent from the room during the first two picks. Thus, he does not know which doors have been selected by A or B. Door #3 is ajar, and C does understand that M opened this door, in accordance with the rules, after A's selection was made.
Remove door #3, and invite contestant D to the stage, like C, D was out of the room during the earlier proceedings, but unlike C, he doesn't not know the game. We simply give D a choice between the two remaining doors.
At this point, the doors are opened. To keep the math simple, assume that the prize is not shared - each player gets full value if they selected the winning door.
The fun part of all this, of course, is that D chooses the same door as A 50% of the time, and D chooses the winning door 50% of the time (assuming D's choice is uniformly distributed).
June 7, 2003 12:21 PM
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After D choses, A is given the option of keeping his original pick or choosing D's pick. Should A change? Does it matter if A doesn't know which door D has selected?
This is equivalent to the proposition "should A switch doors?", scaled by the frequency that D has chosen the other door.
How much do you have to pay ( or charge ) A for the priviledge of switching to D's door? $0 in the case what A knows D chose the same door, $V (sign deliberately left ambiguous) when A knows D chose the other door, $V/n when A knows only that D would choose the other door one time in n (n is certainly positive, but need not be an integer).
Comment by: Danil June 7,2003