You Can Prove Whatever You Want about Streaks and Trends

1 November 2004

By Alan Krigman

If you understand coin flipping, you have a good grasp on the essence of gambling, transcending the rules of individual games. Take the idea of streaks. Or -- if you prefer, trends, patterns, or charting -- which are ultimately variations on a common theme. For instance, say you've watched a coin being flipped 10 times, with the first outcome being Heads and the next nine Tails. Assuming the coin to be unbiased and the flipper not superhumanly skilled, the reality is that the chances of Heads or Tails on the next flip are 50 percent each. The previous flips are irrelevant.

However, solid citizens often believe otherwise. Depending on the book or system they bought, or on the sessions that stick out in their minds, many would argue that the law of averages somehow drives series of events into balance so Heads are now "due" and are therefore favored. Perhaps equally as many would put their money on Tails, thinking the trend likely to continue. The notion of Heads being due, "the gamblers' fallacy," is an incorrect interpretation of probability theory. The concept of streaks continuing is, sad to say, ignorance of the laws of the universe.

Given the original 10 flips, after another 10 trials, probability theory actually predicts six heads and 14 Tails and not 10 each. This, because the next 10 flips are expected to yield 5 of each. Analogously, 90 flips later, the theoretical numbers are 46 Heads and 54 Tails, since 45 of each are expected.

The term, "expected," however is meant in a statistical sense, something akin to "average." It doesn't connote a reasonably certain eventuality. Still, picture what would happen were the expected numbers to materialize. The absolute offsets would still equal eight. This was the difference, 9 - 1 = 8, in the first 10. And, it's the expected difference, 14 - 6 = 8 or 54 - 46 = 8 after totals of 20 and 100 flips, respectively. What changes is the percentage offset. With the 10 flips, proportions of Heads and Tails were wide apart at 1/10 and 9/10 -- 10 and 90 percent. After 20 total flips, the theoretical proportions are 6/20 and 14/20 -- 30 and 70 percent. And when 100 flips are history, the expected proportions are 46/100 and 54/100 or 46 and 54 percent. So values retain the same separation but percentages converge.

To see how this might work in practice, pretend that a gambler watched 10 flips of a coin yield one Heads and nine Tails then jumped into the action. The accompanying table indicates what happened in a dozen different simulated sessions after 10, 90, 990, and 9,990 additional flips. Note that if the simulations were run again, the figures would change -- the purpose of the exercise being to illustrate how tallies vary, rather than values actually to be anticipated in any specific situations.

Numbers of Heads and Tails in Simulated Games of Various Lengths
after one Head and Nine Tails in the First 10 Trials

20 flips			100 flips		1,000 flips		10,000 flips
trial	Heads	Tails	Heads	Tails	Heads	Tails	Heads	Tails
1	5	15	46	54	525	475	5,006	4,994
2	8	12	57	43	504	496	4,886	5,114
3	7	13	44	56	480	520	5,107	4,983
4	7	13	44	56	475	525	4,974	5,026
5	8	12	46	54	487	513	5,041	4,959
6	5	15	53	47	502	498	4,983	5,017
7	8	12	54	46	504	496	5,037	4,963
8	5	15	36	64	493	507	4,950	5,050
9	8	12	44	56	513	487	5,030	4,970
10	5	15	49	51	505	495	4,923	5,077
11	6	14	45	55	472	528	5,002	4,998
12	5	15	42	58	497	503	5,002	4,998

The table shows that the gap between Heads and Tails can widen, narrow, or reverse. And even in what might be considered the "long run" of 10,000 total spins, decisions don't necessarily home in precisely on the expected 4,996 and 5,004. The data in the table also suggest how easy it would be to reach erroneous general conclusions based on what happens in particular games. For example, after 10 additional flips, the gap between results widens in trials 1, 6, 8, and 9; this might wrongly reinforce the beliefs of those who wanted to think that the first 10 flips showed a trend favoring Tails. On the other hand, after 990 further flips, frequencies of the outcomes reverse in trials 1, 2, 6, 7, 9, and 10; this might be taken to falsely infer that after 10 flips heavily biased toward Tails, the law of averages would compensate with a greater propensity for Heads. All reminiscent of the rhyme by the punter's poet, Sumner A Ingmark:

Beware temptation to embrace,
A theory proved by just one case.