50/50, from Duncan Coker

December 10, 2012 |

Jeff's coin proposition bet illustrates a nice lesson for me when applied to trading. That is, even if probability is favorable, there can and will be streaks against. So, there needs sufficient N and staying power for probability to work in trading. So all the seasonal or studies that trade once or twice a year probably don't have a statistical edge.

The inverse lesson is that sometimes it is good not to trade when the probability is not in favorable, as in never take a proposition bet against a Florida surfer with a low handicap, (humor intended).

Jim Sogi writes:

I read that in a sample of 10^10 binomial chances, there can be a run of a 1 million 1's.

The idea that in an infinite random time series every possibility will occur, such as the history of the earth, kind of worries me. There seem to be laws of nature, but are they? Will they change? Do they?

Ralph Vince writes:

James,

Yes, and it is man's innate ability to asses such probabilities (and hence, the fallacy of Huygens and Pascal — that risks should be assessed based on mathematical expectation) that is the most fascinating thing about the entire story of evolution (again, to me).

Why do you get on an airplane when it can crash? Why do you get in your car and go out to buy a quart of milk? We have evolved over eons to pursue often time-critical rewards on a risk-laden planet — it IS how we operate or we would be still cowering agoraphobically in the shadows of a primeval world. This notion fascinated me (and the reason I wrote a book on it in 2011), and the more I dove into it, the more I saw that the answer to it — i.e . the fundamental equations we posses innately for assessing risk, pertains to all other mathematical decision (game theory is rife with concepts that are tuned to the Huygens/Pascal model, not our innate model) and ought to be reassessed under the lens of our superior, realistic model (and yes, it is superior, or we would all be looking for termites to eat up in a tree some place.

Leo Jia writes:

Ralph,

Your notion about man's innate ability to assess probabilities is fascinating to me. I hope to read your new book soon (I presume it is Risk-Opportunity Analysis.)

It is clearly phenomenal that the human species was able to advance over other species. It is not as clear though whether it was man's special innate ability that made man evolve or it was the evolution process that gave man the innate abilities. Regardless of whatever came first, I think many of man's innate abilities that exist today were largely fostered by the evolution process. While this was wonderful, it is perhaps also very discomforting to learn that many of our innate abilities were more meant for the environment of the wild, not really for the modern times as the modern couple hundred years is far too short in evolution terms. It begs the question of what of the very innate abilities are really useful and what are not. Whether we realize what abilities we have or not perhaps is not a big issue as we naturally use them in life. It does become more important for us to know what of our innate abilities are actually harmful to ourselves today.

I did a test. It went like this:

1) toss a coin 10 times,
2) if there is 5 heads then add 1 to a record do the above 2 steps 1 million times.

The chance that in ten tosses one gets exactly 5 heads and 5 tails is 24.5539%.

To be more comprehensive with the test results:

4 heads and 6 tails: 20.4194%

6 heads and 4 tails: 20.5125%

3 heads and 7 tails: 11.7019%

7 heads and 3 tails: 11.7010%

2 heads and 8 tails: 4.4018%

8 heads and 2 tails: 4.4145%

1 heads and 9 tails: 0.9783%

9 heads and 1 tails: 0.9830%

0 heads and 10 tails: 0.1004%

10 heads and 0 tails: 0.0968%

Easan Katir writes:

Thank you, gentlemen. This is good info to ponder and apply to trading. For my part, I found a shiny Lincoln-cent and spun it 10 times. Result: 7 heads.

Jeff Watson writes:

But there is also another trick of spinning a coin very fast, get down to coin level on the table and observe carefully, and if you get a blurring image of tails, call tails…same thing if you see heads, call heads. Since the coin spins at a slight angle, the side that you can see the image will be what lands.

Gentlemen,

As far as coin tosses and trading — and this may be redundant information to many of you — to me, personally (in my sciatica and failing vision nowadays) I find the largest implication pertains to the nature of the equity curve and expectations, and the deceiving nature of randomness.

We know if we plot out the equity curve of consecutive coin tosses (with heads +1, and tails, -1, say) and we plot this out, we can then draw bands around the mean expected value (0 in this case) of standard deviations. Thus, we can draw a one standard deviation band above and below.

Such a band will be parabolic, like a parabola resting on its side, rightward-facing, opeining up as time or trades or plays go by. That is, the upper band will always be ever increasing albeit at an ever decreasing rate. Thus. to be ahead of the expectation by play number X to the tune of 1 standard deviation, is below being ahead of the expectation by play X+1 or X + N where N is any positive number.

Couple this now with the Second Arc Sine Law*, which pertains to such randomly-generated equity streams and tells us (the essence of The Second Arc Sine Law) that we would expect both the peak and nadir of equity stream to occur least likely towards the center (time-wise) and most likely near the start or finish of such a stream.

These two principles, take together, warn us that in a stream of randomly-generated outcomes (coin tosses, or trading if/when the outcomes occur with randomness) we should expect the rightmost endpoint to be at or near the very top (or bottom) of the entire equity run, deluding us into conclusions, "This works!" or "This fails," that have no basis in a causal existence, but are merely the artefacts of randomness.

*The First Arc Sine Law buttresses this further, this law being that we should expect the ratio of the cumulative equity line (comprised of X number of plays) least likely to be above the expectation X/2 number of times, and most likely to be above or below X or ) number of times — the same Arc Sine distribution as the Second Law. Thus, say, if I toss a coin ten times, it has an expectation of 0 (given the caveats mentioned in this thread!) and I would expect with highest probability that ten of those tosses see the cumulative equity line above (or below) the expectation line of 0 and with the least probability, see 50% of them above and below the expectation (0) line.