Two Goats and a Sportscar - Solution
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Two Goats and a Sportscar - Solution |
She should switch doors. This doubles her chances of winning the sportscar.
Surprised? This is a rather subtle puzzle and a tricky one to get your head around. In fact, you'll (probably) be totally convinced that the explanation below is complete nonsense. But I'm (fairly) sure it's right...
Since there are two goats and one car, the chance of her picking the sportscar with her first choice is 1/3, and of picking a goat is 2/3. This is straightforward probability.
If she sticks, this doesn't alter her chance of winning the sportscar (ie 1/3) [A] or goat (ie 2/3).
However, notice that, now that one of the goats has been revealed, if she switches she'll always get the opposite of what she originally picked:
if she originally picked the sportscar, then switching means she gets a goat (in fact, this would have been the case even if the goat hadn't been revealed);
if she originally picked a goat, then switching means that she gets the sportscar (since the one remaining goat has been revealed, so the only thing that can be behind the remaining closed door must be the sportscar).
So, since switching means that she gets the opposite of her original choice, and the probabilities of her original choice were 1/3 for the sportscar and 2/3 for a goat, then the probabilities after switching are 1/3 for a goat and 2/3 for the sportscar [B].
So, if she sticks she has a 1/3 chance of winning the sportscar (see [A]), but if she switches she has a 2/3 chance of winning the sportscar (see [B]).
In other words, she doubles her chances of winning the sportscar (from 1/3 to 2/3) by switching.
The solution depends crucially on the order that things happen, in particular the fact that the woman picks a door before the host reveals one of the goats. It's an example of Bayesian logic, as described very expertly in Chapter Three of Ian Stewart's The Magical Maze.
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© 2002 Ian Hadden |