Wait, here’s another popular theory about the chance of things happening in series: the longer you flip a coin without heads coming up, the more likely it is that heads will be next. So you should increase your bet with every turn. This theory is called “Gambler’s Ruin”. It's wrong because it confuses the odds of a single event (the third penny coming up heads) with the odds of a series (all three coming up heads).

The basic law of probability is that the likelihood of a series of outcomes (all heads) is equal to the likelihood of the first outcome, times the likelihood of the second, times the likelihood of the third, and so on.  If there is a .5 chance of the first coin being heads, a .5 chance of the second coin being heads, and a .5 chance of the third coin being heads, the likelihood of all three coming up heads is therefore .5 x .5 x .5 or .125.  1 in 8. Since in this case you get paid for all heads or all tails, you will get a dime once in every four tosses, and you will lose a nickel on each of the other three. You figure it out.

Although he is known primarily for his ideas about gravity and his use of the telescope, Galileo Galilei (1564-1642) the Italian mathematician and philosopher, also wrote some of the earliest papers on probability in games of chance, and established much of the basis for our modern study of statistics. "Philosophy", he wrote, "is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth." Il Saggiatore, 1623