Do you think playing two losing games can result in a win streak? Parrando’s paradox proves this can happen in a very simple way.

The paradox is illustrated by two games played with coins weighted on one side so that they will not fall evenly by chance to heads or tails.

In game A, a player tosses a single loaded coin and bets on each throw. The probability of winning is less than half. In game B, a player tosses one of two loaded coins with a simple rule added. He plays Coin 1 if his money is a multiple of a particular whole number, like three.

If his money cannot be divided by the number three, he plays the Coin 2. In this setup, the second will be played more often than the first.

Both are loaded, one to lose badly and one to win slightly, with the upshot being that anyone playing this game will eventually lose all his money.

“Sure enough,” Dr. Abbott said, when a person plays either game 100 times, all money taken to the gambling table is lost. But when the games are alternated — playing A twice and B twice for 100 times — money is not lost.

It accumulates into big winnings. Even more surprising, he said, when game A and B are played randomly, with no order in the alternating sequence, winnings also go up and up.

This is Parrando’s paradox.

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Today marks the 306th birthday of famous Swiss mathematician, Leonhard Euler (1707-1783), as you might have noticed from the unique Google logo. Euler made a significant number of discoveries in various fields, from infinitesimal calculus to graph theory. He is well known for introducing the notion of a mathematical function and his works take up between 60-80 quarto volumes. Here are just some of his accomplishments.

Euler’s formula

Euler’s formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely large region), then

v − e + f = 2.

Seven Bridges of Königsberg

The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory and prefigured the idea of topology.

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. Euler proved that the problem has no solution. There could be no non-retracing the bridges. The difficulty was the development of a technique of analysis and of subsequent tests that established this assertion with mathematical rigor.

Euler’s identity

In analytical mathematics, Euler’s identity (also known as Euler’s equation), named for the Swiss mathematician Leonhard Euler, is the equality

where
e is Euler’s number, the base of natural logarithms,
i is the imaginary unit, which satisfies i2 = −1, and
π is pi, the ratio of the circumference of a circle to its diameter.

Anscombe’s quartet comprises four datasets that have nearly identical simple statistical properties, yet appear very different when graphed. Each dataset consists of eleven (x,y) points. They were constructed in 1973 by the statistician Francis Anscombe to demonstrate both the importance of graphing data before analysing it and the effect of outliers on statistical properties.

The first scatter plot (top left) appears to be a simple linear relationship, corresponding to two variables correlated and following the assumption of normality. The second graph (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear, and the Pearson correlation coefficient is not relevant. In the third graph (bottom left), the distribution is linear, but with a different regression line, which is offset by the one outlier which exerts enough influence to alter the regression line and lower the correlation coefficient from 1 to 0.816. Finally, the fourth graph (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

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