It's a Puzzlement

Parrondo's Paradox

Some of you may have seen this in the news recently. Consider three games, each of which involves flipping coins. You start with some amount of capital, say $1,000. Your capital goes up by $1 if the coin comes up heads, and goes down by $1 if the coin comes up tails.

Game A—Use a coin that comes up heads with probability 0.495.

Game B—Use two coins, one that comes up heads with probability 0.095, and another that comes up heads with probability 0.745. If your capital is a multiple of 3, use the first coin, otherwise use the second coin.

Game C—Flip a fair coin to determine whether to play Game A or Game B (play one round of whichever game is selected, and then flip the fair coin again, and so on).

The puzzlement is to determine what is likely to happen to your capital if you play any of these games some number of times. For example, try each game 100,000 times, starting with $1,000 capital. Explain what happens. (Note that I used the word "likely" in the first sentence because statistics means never having to say you're certain.)

Exact Ranking

In this puzzlement, David Bickerstaff was trying to guess a ranking. His first guess was A-B-C-D-E. He was informed that he had each person out of her true position, and not one in his ranking followed her immediate predecessor. He then asked if it were D-A-E-C-B. He was told that he had two in the correct position and two correctly following her immediate predecessor. David was then able to determine the correct ranking. What is it?

The correct ranking is E-D-A-C-B. Mike Fusco's solution is essentially as follows. The two that are in the correct position must be consecutive otherwise there is no way to have two correctly following their immediate predecessor without having at least three in their correct position. A little testing shows that the two in their correct position are C and B. Then either A follows D or E follows A. If the latter, then neither possible ranking meets both conditions. So it must be that A follows D and the solution is as given.

Solutions were also sent in by Martin Adler, Michael Belfatti, Mary Ellen Cardascic, Al Commodore, Brett Gissel, John Herder, Charles Hewitt, Paul Ivanovskis, Ira Kaplan, Frank Karlinski, Mark Kertzner, Alex Kozmin, Richard Newell, Nick Pastor, Dan Post, Evan Spiegel, Walter Wright, Christopher Yaure, and Anthony C. Yoder.