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The Sloppy Monty Hall Puzzle

Why a Little Learning can be a Dangerous Thing

This morning's "A Point of View" on Radio 4 began with a presentation of what is widely known as the Monty Hall Problem.  There's a transcript here:

If you already know the puzzle and are thinking "Yeah yeah, switch the doors...!" then bear with me - there's a twist that you won't have encountered before. 

If you haven't heard the puzzle, here's a quick summary. In a gameshow there are three doors, behind one of which is a star prize, while the other two have nothing behind them.  You choose a door (door A, for example).  The host then opens another door (B, say) and shows there is nothing behind it.  You are now offered the chance to swap to Door C, or to stick with the door you were first given (A).  What should you do?

It's now "common knowledge" that you should swap doors, because in doing so you increase your chance of being right from 1/3 to 2/3.  I've tested this on lots of sixth form mathematicians, and the majority "know" that you should swap.

What those who automatically say "swap" don't realise is that the decision on whether you should swap or not depends on what the strategy of the gameshow host is.  The assumption on today's radio item was that the gameshow host KNOWS which of the unchosen doors have nothing behind them, and that whatever you choose, he will always then select a door with nothing behind it and will open that.  In doing so, he has given you no useful information, so your chance of being right with your first choice (Door A in the example above) is still only 1/3.

But why should we assume that the gameshow host is using this rule, unless we are explicitly told that he is doing so?

These days, with mathematical audiences I usually perform the Monty Hall game as follows:  I tell my volunteer that they will be given a chance to pick a door, and that after they have chosen a door, I will be giving them a chance to swap . 

If they choose the door with the prize behind it, I then reveal an empty door and ask them if they want to swap.  Invariably they do, because that's what you are "supposed" to do.  So they lose!

If, however, they don't pick the prize door first time, I then ask them to pick one of the other two doors, telling them that it will now be removed from the choice of three.  Having done that, they are then allowed to either stick or swap to the remaining door (as I promised).  In this case, there is a 50-50 chance that the door they excluded was the one with the star prize, so it doesn't matter if they stick or swap, they won't win the star prize.  Rarely does anybody spot that I have subtly changed the normal Monty Hall rules.

Needless to say, when playing with mathematicians in this way, I win Monty Hall most of the time.

So watch out for sloppy presentations of the Monty Hall puzzle...and remember that a little learning can be a dangerous thing.