# Hint to Riddle #78: Secret Santa Maths

78. Secret Santa is a Western Christmas tradition in which members of a group or community are randomly assigned a person to whom they anonymously give a gift.

The rules for determining who buys for whom are as follows. All the participants put their names in a hat. Each player will draw a name in turn. If they should draw their own name they will return it to the hat and draw another. If the last person to draw draws their own name the game is void and everyone starts again. Otherwise people will not return the name to the hat and will buy a present for the person whose name they have drawn.

Adam, Bob and Carol are to play a game of Secret Santa. They will draw in alphabetical order. What are the odds that Adam buys for Carol?
So it looks like the answer is 50/50, a half. The first move of the game is A pulling a name out the hat. If he pulls his own out he puts it back but other wise he has a 50/50 chance of picking B or C. So where's the problem.

We will cover the maths, but in principle the problem is this. If a chooses B or C the odds of the game completing successfully are not the same. Intuitively from the criteria for voiding the game 'If the last person to draw draws their own name the game is void and everyone starts again.' we can guess that this is more likely to happen if the last person, C, still has their name in the hat.

Draw a flow chart or a decision tree and work out the probabilities of each avenue.

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