Roger,
Great article. I get it but its still hard to believe. Seems to me that once the one door is eliminated the odds are 50/50 with either door. Another way to look at it would be what if you had two people (A & B) selecting different doors at the same time. Then the third door was revealed which had the goat. Who is more likely to win A or B. In that case it would be the same.
Oh well thanks for the ariticle.
I'll try to explain it in words, without going through the math.
When you first select a door, there is a one-third chance that you picked the correct door, and a two-thirds chance the correct door is one of the two you did not pick.
Now, let's imagine a slight variation of the game. The rules are the same, except that when Monty shows you a door he also has no idea what is behind the door he picks. If Monty's door has the car, you lose and the game is over. If Monty's door has a goat, you have the choice of keeping your door or selecting another.
Note that the only difference between this and the "real" game is that Monty does not know what is behind his door. Since Monty doesn't know where the car is, one-third of the time he picks the car and you lose.
The other two-thirds of the time - when Monty's door has a goat - the car is equally likely to be behind either door.. In that case it makes no difference if you switch. Note that in this variation of the game your odds of winning are one-third no matter what strategy you employ. One third of the time you lose when Monty opens his door and you aren't even given a chance to switch. Of the remaining two-thirds, you win half of the time.
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But Monty doesn't play the game that way. Monty knows what door has the car, and when he picks a door he
never picks a door that has a car. IOW - Monty eliminates the one-third of outcomes where you lose without having a chance to pick a door. (If you're a craps player, it's like playing a craps game in which 2, 3 or 12 aren't craps - you can never lose on the come out roll.)
Think about how that changes the game. Go back to when you first picked ad door. At that point there was a two-thirds chance that you picked the wrong door. That situation remains. When Monty opens his door, you now know that:
- There is a two-thirds chance that the car is behind one of the doors you didn't pick.
- The car is not behind the door that Monty picked - which means that there is a two-thirds chance the car is behind the door Monty did not pick.
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I think that what hangs up many people on the Monty Hall situation is that they don't appreciate the significance of the fact that Monty does not open a door randomly. In the first variant I laid out, Monty does open a door randomly; in that case door switching doesn't make any difference. But when Monty doesn't select a door at random, the odds change.
