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Three Boxes and a Ruby

Puzzle

Alice places three identical boxes on a table. She has concealed a precious ruby in one of them. The other two boxes are empty. Bob is allowed to pick one of the boxes. Among the two boxes remaining on the table, at least one is empty. Alice must then remove one empty box from the table. Finally, Bob is allowed to open either the box he picked, or the box lying on the table. If he opens the box with the ruby, he gets a kiss from Alice (which he values more than the ruby, of course). What should Bob do?

Source

The problem is popularly known as the Monty Hall Problem.

Solution

If Bob switches his choice, he wins with probability 2/3. There's a YouTube video explaining the solution! To get better intuition, it helps to consider a slightly different problem with 100 boxes with one box containing a ruby. Bob then picks one of the boxes at random. At least 98 of the remaining boxes are empty — these are removed by Alice. So now, we are left with two boxes: should Bob switch? Indeed, for he wins with probability 99/100 if he switches!

Followup

One family has two boys and one girl. Another family has two boys and two girls. If we were to pick two children from a family, uniformly at random, then for which family are the chances of picking two children of the same gender more? A generalization: consider two jars. The first jar has n red and n blue marbles. The second jar has n red and n-1 blue marbles. Which jar has larger probability of drawing marbles with identical color?

Previous Puzzle: Two Dice = Nine Cards

Alice has two standard dice with labels 1 thru 6. When she rolls them and adds their labels, she gets a distribution over integers in [2, 12]. Bob has nine cards, each labeled with some real number. When Bob chooses two cards (without replacement) and adds their labels, he gets exactly the same distribution over integers in [2, 12] as Alice gets by rolling her dice. What are the labels on Bob's nine cards?

N women stand in a queue to take seats in an auditorium. Seating is pre-assigned. However, the first woman is an absent-minded professor who chooses any of the N seats at random. Subsequent women in the queue behave as follows: if the seat assigned to her is available, she takes it. Otherwise, she chooses an unoccupied seat at random. What are the chances that the last woman in the queue shall get the seat assigned to her?

12 Sep 2008
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