Double threat
Suppose you are in a dungeon. There are 50 doors numbered 1 to 50. Only one of them leads you out of the dungeon but behind each of the other 49 is an evil fire-breathing dragon. The doors are watched over by one guard whom you can only ask yes/no questions to determine the correct door.
But here's a catch: every time you ask a question, the guard, who has 10 children (9 boys and a girl), will randomly write the name of one of his 10 children. If he writes any of his son's names, you survive. But should he write his daughter's name, an alarm will ring and all the dragons will break through the doors.
What is your chance of survival? Give your answer as a decimal between 0 and 1.
Assumptions
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Encountering a dragon means certain death.
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The guard always tells the truth.
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You optimise your strategy to maximise your chance of survival.
The answer is 0.531.
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1 solution
The best type of questions to ask would be those where if the guard answers yes/no, the number of possible doors gets reduced by half. You should start with 'Is the correct door number greater than 25?' followed by 'Is the correct door number greater than 12/37?' How many times do you need to so this? Since 2 5 < 5 0 < 2 6 , you would need 6 such questions to pinpoint the correct door. Every time you ask, you have a 0.9 chance of survival. Hence 0 . 9 6 = 0 . 5 3 1 . Note that this is still more than half so do not consider guessing between two or more doors where your chance of survival is at most 0.5.