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Saturday, July 26, 2008

The Prince and the Pearls

Long ago, a young Chinese prince wanted to marry a Mandarin's daughter. The Mandarin decided to test the prince. He gave the prince two empty, porcelain vases, 100 white pearls, and 100 black pearls. "You must put all the pearls in the vases", he told the prince. "After this, I will call my daughter from the room next door. She will take a random pearl from one of the two vases. If this pearl is a black one, you are allowed to marry my daughter."

First Question: What was the best way in which the prince could divide the pearls over the vases?

Second Question: You have three vases: one vase containing two white pearls, one vase containing one white and one black pearl, and one vase containing two black pearls. From one of these vases, a pearl is taken. This pearl turns out to be white. What is the probability that the other pearl in the same vase is also white?

Third Question: You have ten vases. Five of the vases contain a white pearl and four of the vases contain a black pearl (note that a vase may contain both a white and a black pearl!). You randomly select one of the ten vases. What is the probability that the vase you chose is empty?


First Answer: The best way is to put one black pearl in the first vase, and all other pearls in the second vase. Then, the probability of grabbing a black pearl from the first vase is 1, and the probability of grabbing a black pearl from the second vase is 99/199. The total probability of grabbing a black pearl is 0.5 × 1 + 0.5 × 99/199 = 298/398 (approximately 74.9%).

Second Answer: There are three pearls that can be the white pearl that was taken from the chosen vase:

* The first pearl from the vase with two white pearls: in this case, the other pearl is also white.
* The second pearl from the vase with two white pearls: in this case, the other pearl is also white.
* The white pearl from the vase with one white and one black pearl: in this case, the other pearl is black.

The probability that the other pearl in the same vase is also white, is therefore 2/3.

Third Answer: The probability that the chosen vase does not contain a white pearl is (10-5)/10 = 1/2. The probability that the chosen vase does not contain a black pearl is (10-4)/10 = 3/5. The probability that the chosen vase does not contain any pearl is therefore 1/2 × 3/5 = 3/10 (which is 30%).

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