Puzzler Answer 2/21/97: Pearls

Ray:
You know Valentine's Day was recently upon us and I had to get my wife a present and thought I'd do something unique this time. For years now she's been asking for a string of black pearls. They're very rare and the only place they're found is off the waters of the Seychelles--those islands in the Indian Ocean where they just did the latest edition of the Sports Illustrated swimsuit issue. That's the only reason I knew that. These pearls are brought up from the depths by divers who go down 200 feet with a knife clenched in their teeth and open up the oysters and if there is one of these pearls down there they, of course, bring it to the surface, and if not, they eat the oysters...they get the bends and they keep asking for divers. Anyway, my wife asked for these pearls and I said, "Geez, these are a lot of money." So I proposed a little game.

Tom:
Instead of a boat payment.

Ray:
I get 50 of these black pearls and I put them in a cigar box. And I get 50 faux white pearls. So, I've got 50 black pearls from the Seychelles and then I have 50 white pearls in another cigar box. And I tell her, "Look I'm going to put these cigar boxes in front of you, you will be blindfolded and you will instruct me to open one or the other, either A or 2. You will then pick a pearl out and if it's a black one, you get the black pearls, and if it's a white one, you get the cheap pearls that I had intended to buy you in the first place. So it's obvious, since there are 50 of each pearl, her chances are 50/50. But she can mix them up. She can put all the pearls in one box, she can put half the white ones in one box and half the black ones in the same box, so she has 25 of each color in one box. Is there anyway she can mix up these pearls to improve her chances beyond 50/50? That's the question.

Tom:
She can do anything she wants with the pearls.

Ray:
She has to use all the pearls and she has to put them in these two boxes. Now clearly they're already 50 in one box and 50 in another. Knowing a little about probability I know that her chances of getting a black one are 1 in 2. Can she improve those odds beyond 50/50? And the answer is -- yes. Because she can take all the pearls except for one black one and put them in box A.

Tom:
Oh! So she takes all the black pearls, except for one, and dumps them in the other box.

Ray:
In the remaining box she has one black pearl. So she should pick that box at random She will certainly get a black pearl, and if she picks the other box her chances are still 50/50 or close to it.

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