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Summer Maths Puzzles - Thursday 15 August - A Prize and 2 Goats

A Prize and 2 Goats

This morning the children were having an argument.  This reminded the parents of the Monty Hall problem, infamous for having caused major arguments amongst some leading mathematicians.  It goes like this:

In a game show there are three doors, two of which have a goat behind them and one of which has a major prize.  The contestant chooses a door but doesn’t open it.  The game show host then opens one of the other two doors to reveal a goat before asking the contestant whether they want to stick with their original choice or swap and open the other door.

So the question is, should the contestant stick with their original choice and open that door and win the prize behind it, or should they swap and open the remaining door and win the prize behind that.  The (somewhat surprising) answer, probabilistically speaking, is that they should swap: they are twice as likely to win the prize if they swap than if they don’t.

Having explained the Monty Hall problem to the children, the parents asked a more generalised question: suppose there are n doors and g goats, should they still swap?  What is the probability that they win if they swap?  Is there a relationship between n and g that determines whether they should or shouldn’t swap?

Can you help the children answer this question?

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons