Although Anything Can Happen, Something Else Usually Does

Someone gives you and three friends each a coin and asks you to do 10 flips, jotting down the outcomes with "0" for tails and "1" for heads. You get 0110001011, Elvira's flips are 1000110110, Flossie's are 1010101010, and Roscoe ends up with 1111111111.

Your flips and Elvira's are unremarkable. Each has five heads and five tails, obeying your idea of the law of averages. No patterns are obvious. And, if you perform a standard statistical test for randomness by tallying numbers of "runs" of various lengths, both sequences would pass muster with six each: three of length one, two of length two, and one of length three.

Flossie's and Roscoe's records are surprising, to say the least. Flossie's meets your intuition of probability with a 50-50 split of heads and tails; but it has a pattern, and fails statistically with 10 runs of length one. Roscoe's meets neither probabilistic nor statistical yardsticks, having too many heads, an evident pattern, and too few runs. These conditions would lead most observers to suspect that something strange was taking place.

Here's another wrinkle. Before sending you off to do your flips, the Magister Ludi secretively scribbles something on a slip of paper and drops it into an empty box. After each of you reveals your results, you open the box and read what's written: "Elvira, 1000110110." Now, Elvira's results are suspicious, even though they meet the normal criteria for random flips of a fair coin.

This mental experiment highlights two phenomena of a type that mislead many solid citizens in their casino gambling endeavors.

First, when you flip a coin 10 times, you'll get one of 1024 possible combinations of heads and tails. Assuming the coin is unbiased, each combination is as likely as any other. Your 0110001011 is no more or less apt to occur than 1010101010 or 1111111111. Although yours doesn't seem peculiar in any way, it's still just one out of 1024 equally probable combinations and therefore has a chance of 1/1024 - just like any of the others.

If this seems counterintuitive, consider what you think about Elvira's series before and after seeing the paper in the box. Nothing seemed bizarre until the prediction proved to be correct. Then you might wonder about the probability this would happen solely by luck; the answer would be 1/1024, the same as for any other specific combination. The same as 1010101010 or 1111111111.

Second, when you flip a coin 10 times, the "expected" number of heads and tails is five each. But "expected" is a statistical term, and by no means tells what will actually happen. In fact, based purely on the ways any given number of heads and tails can be arranged, you can find the following probabilities:
Chance of various numbers of heads in 10 flips

no of heads	ways	probability
0	1	0.098%
1	10	0.977%
2	45	4.394%
3	120	11.719%
4	210	20.508%
5	252	24.609%
6	210	20.508%
7	120	11.719%
8	45	4.395%
9	10	0.9775%
10	1	0.098%

The list shows that, in 10 flips, five heads can occur most often - 252 out of 1024 ways. This is why five is the "expected" value. But four or six heads aren't all that unlikely, with 210 ways each. And so on, along the line. There's nothing mystical behind these figures. Just a matter of counting arrangements. Zero heads can occur one way: 0000000000. One can occur 10 ways: 1000000000, 0100000000, 0010000000, 0001000000, 0000100000, 0000010000, 0000001000, 0000000100, 0000000010, and 0000000001. You can try it yourself with other numbers of heads, but the enumerations get messy so you might prefer to take me at my word.

What does this say about a large class of gambling systems? That the unexpected can occur, and sometimes does. But not usually. The poet, Sumner A Ingmark, resolved the riddle rhymingly:

Don't plan on aberrances,
They are rare occurrences.