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March bracket madness

The last installment of Chance News, Warren Buffett's challenge on the NCAA basketball tournament.

We received the following news updates from Jim Greenwood and Margaret Cibes

• Mathematicians are hoping their calculations add up to the perfect bracket, by Mary Pilon, New York Times, 15 March 2014

• In N.C.A.A. tournament bracket, mathematician outdoes Matildas, by Joe Drape, New York Times, 24 March 2014

These articles describe efforts by aDavidson College math professor, Tim Chartier, who used an algorithm based on applied linear algebra to make bracket predictions. Last year, three of Chartier's students scored in the 96th to 99th percentiles among thousands of entries in ESPN's bracket contest.

A pair of probability puzzles

[A coin problem]

Consider this simple game: flip a fair coin twice. You win if you get two heads, and lose otherwise. It’s not hard to calculate that the chances of winning are 1/4.

Your challenge is to design a game, using only a fair coin, that you have a 1/3 chance of winning.

Continues "And here is my recipe for getting the most out of this problem: if you can solve it, do not stop with one answer. Rather, see how many answers you can come up with. I’ve posed this problem to many people, and I continue to hear novel solutions."

A large urn
by Gary Antonik, Numberplay blog, New York Times, 24 March 2014

There are 600 black marbles and 400 white marbles mixed well in a large urn. You draw marbles one by one at random without replacement until you take out all the marbles of one of the colors. What is the probability that at least one white marble will be left in the urn?

Bonus: On average, how many marbles will be left in the urn?

Submitted by Bill Peterson