I feel better now.
Hey, if anyone thinks they're being alive on earth is just by chance, trying
estimating the odds of your parents getting together AND a specific cell
of your Dad meeting a specific cell of your Mom. After you have that estimate,
(ha ha) go back to their parents and do it again. (S.S.)
Math quotes time:
If you think dogs can't count, try putting three dog biscuits in your pocket
and then giving Fido only two of them.
Pure mathematics is, in its way, the poetry of logical ideas. ~Albert Einstein
Black holes result from God dividing the universe by zero. ~Author Unknown
Arithmetic is where numbers fly like pigeons in and out of your head. ~Carl Sandburg, "Arithmetic"
If equations are trains threading the landscape of numbers, then no train stops at pi. ~Richard Preston
Can you do Division? Divide a loaf by a knife - what's the answer to that?
~Lewis Carroll, Through the Looking Glass
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. ~Albert Einstein, Sidelights on Relativity
If there is a God, he's a great mathematician. ~Paul Dirac
One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers... ~Heinrich Hertz
In the binary system we count on our fists instead of on our fingers. ~Author Unknown
The laws of nature are but the mathematical thoughts of God. ~Euclid
Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere - Archimedes did it in a bathtub. ~Richard J. Trudeau, Dots and Lines
If a coin is flipped it has a 50:50 probability of coming up heads.
Someone flips a fair coin 9 times and comes up tails each time.
(This is NOT part of a larger number of flips, like 100 or 1000.)
The coin doesn't know about probabilities so it's not due, so to
speak, to be heads at the 10th flip.
This someone, flips the coin the 10th time. I don't get why the
odds of it coming up heads are still 50:50. Is it because of when
we ask about the probability of it coming up heads?
If we ask before the first flip...
What is the probability of 10 tails in a row? Then the odds are
greatly against. 2 to the 10th, or whatever.
If we ask after the 9th flip (remember, all previous flips were tails)...
What is the probability of 10 tails in a row? Then the odds are 50:50 ?!
is (1/2)^10 = 1/1024. As for each of the flips the outcome head or tails has the same probability, this is just one out of 1024 possible events, which all have the same probability. For example, the probability for the event
would also be 1/1024, as the probability for each of the 10 coin flips is the same again.
If you ask about the probability for the 10th flip after the result for the first 9 flips is already known , it is 1/2 for both possible outcomes. Note that this is a conditional probability. The condition "9 flips with tails" has itself a probability of (1/2)^9 and you can calculate the total probability for the 10-flip result as
P = (1/2)^9 * (1/2) = (1/2)^10
This multiplication is only allowed if the single flips are independant from each other. And indeed, for a fair coin, the result of the 10 th flip will not depend on the prior results. It is important to understand that notion of stochastic independence.
Application to Trading
The Gaussian distribution which is assumed for the Black-Scholes formula and which is also used to calculate the probability that price will remain within a Bollinger Band using 2 standard deviations, assumes stochastic independance of consecutive changes of price.
This assumption is false, as we can find that consecutive changes in price show a - slightly positive - autocorrelation. The price action on your chart is represented by bars, and the 15:05 bar is not independant from teh 15:00 price bar, because traders are watching price and base their decision to buy or sell on the price action that could be observed before. This is simply feedback.
Whenever feedback is involved, the stochastic independance is lost.
Emotions, greed, fear, avalanches, panics produce typically positive feedback. Arbitrage traders create negative feedback. In both cases you are not allowed to multiply probabilities as for the fair coin flip. Strong feedback loops also invalidate probabilities that were calculated by arbitrageurs such as LTCM. The one-in-ten-thousand-years event happened after 4 years, because the (false) probability calculations were based on Gaussian distributions that did not apply. LTCM collapsed, because it was overleveraged, correlations between the different business models were underestimated and they did not fully understand the arbitrage risk resulting from positive feedback.
The following 2 users say Thank You to Fat Tails for this post:
Ok...so it's (1/2)^10 = 1/1024 for the first idea and (1/2)^10 = 1/1024 for the second, correct?
I can see there is feedback here, but is it random? Some trading systems are profitable on the day in question
and some are not. Some traders see the price going up and buy and others see the price going up and sell.
The price is, at times, going up and down simultaneously, as in the case when trader A using 5 minute bars
sees a higher close and goes long, yet trader B using 15 minute bars sees a lower close and goes short, on
the same instrument.
The following user says Thank You to stephenszpak for this post:
If there is feedback involved, it cannot be random. You would make the distinction between two cases
(1) Usually noise traders enter trades in a random way. Each trader may have a reason, but the overall effect comes close to a random movement as their reasons are arbitrary. If random movements prevail, prices will sit in a congestions or trading range. The price changes can be well approximated by a Gaussian distribution. In this situation you can enter mean reversion trades, if price moves further away from the center of the value area.
(2) Sometimes feedback - either driven by investor sentiment or by algorithms that try to exploit prices moves - may drive prices further away from value. This can be called a bubble, a crash, irrational exuberance, or it can simply be a trend. The trend relies on positive feedback, as one agent buying induces the next agent buying to participate in the price move. When this happens, the actions of the agents are no more independent. Price change does no more follow a Gaussian distribution. If somebody tells you that 95.4% of all price changes will fall into a band (Bollinger Band) with 2 standard deviations around the moving average, this only applies to Gauss distributions. When investor sentiment produces a collective action, the distribution changes, and the statement is no longer true.
If the 5 min trader sees something different than the 15 min trader, there need not be any contradiction. Positive or negative feedback is always temporary, before price finds back to random moves within the newly established value area.
The following user says Thank You to Fat Tails for this post:
I get it. I don't get it. I can see you've thought about this more than I.
There have been a number of mathematicians before the mid 20th century that actually flipped a fair
coin 10000 times or more. (I tried to find their actual recorded data of the flips but just can't.) Many years ago
I think I saw it briefly. Anyway each coin flip in the XX,XXX total is independent of all the others, yet there were
runs of >10 heads in a row and >10 tails in a row. There was no feedback, yet there were trends.