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How advanced mathematics and gaming theory can help you as a trader
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How advanced mathematics and gaming theory can help you as a trader

  #181 (permalink)
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traitor786 View Post
the question should be.

if a coin is fliped 3 times , and 2 heads show up, what are the chances that the next flip comes a head also.

since the last flip has not been made, do we just assume it is an individual flip at 50 50 or do we look at the previous flips and say that it is .5 to the power of 3 ?

In order for it to be a falasy it should go against common sence to some respect and common sence would state that the last flip is not a 50 50.

if you believe the last flip is a 50 50, then all flips are 50 50.

if that the case then why do we even quote the formulae .5 x .5 x .5 ?

maybe im just unclear on what side you are on or maybe im not understanding something here

OK, traitor, let me try this approach. All probabilities that are being discussed in this coin flip example are in the future. If the coin flip has already occurred then the probability of its result is 1.0 which is a certainty.

So if the flips have not been done, then the probability of H H H is 0.5 x 0.5 x 0.5 = 0.125

If the first two flips have been completed and they happen to come out as H H, the probability of another H is 0.5. The fact that two H occurred in the past has no effect on the outcome of the third flip. The gambler's fallacy suggests that a T is more likely on the third flip which is not true.


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  #182 (permalink)
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Fat Tails View Post
I was talking about conditional probabilities of finite subsets.

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  #183 (permalink)
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Lampert View Post
OK, traitor, let me try this approach. All probabilities that are being discussed in this coin flip example are in the future. If the coin flip has already occurred then the probability of its result is 1.0 which is a certainty.

So if the flips have not been done, then the probability of H H H is 0.5 x 0.5 x 0.5 = 0.125

If the first two flips have been completed and they happen to come out as H H, the probability of another H is 0.5. The fact that two H occurred in the past has no effect on the outcome of the third flip. The gambler's fallacy suggests that a T is more likely on the third flip which is not true.


I understand what you are saying. Any flip has a 50 - 50 chance whether it is the first flip or the 10th. It is the laws of gravity airo dynamics and applied force. Nothing to do with math. Its random.

But then in a sequence things change. hhh is 0.125 suggesting that the flip after hhh has a 90% chance of being T. But since it is a flip and all flips have a 50 50 chance we have a conflict.

Does teh coin know if it is in a sequence ?

further all flips to some degree are in a sequence. It could be the first flip of the coins life and it will still be in a sequence as cone can chart all coin flips ever made on any coin and add this one to the data.

or it could be flips of the coins life time. To qualify as a set there is nothing saying that teh flips have to be within 10 secs apart.

if i take 3 coins and do 2 flips on each on and only one comes out HH then i take only that coin to my friends house and and bet him 80% of my money that it will be T will it be a high probability gamble ?

or am i still at 50 50

if i do 1000 coin tosses and end up with a 40-60 heads to tails situation what will happen when i change the coin. will the trend continue ?

can a coin be fliped with out having any previous data?

if the third flip has a .125 chance of being T then by definition the third flip. which can be said to stand on its own will not be 50 50 .

the definition of stand on its own and sequence here are off. and it may be impossible to judge if a flip is on its own aor part of a sequence we have not even seen.

we can calculate all flips world wide ever made as the sequence. we can count flips made this year only, we can calculate flips of quaters only we can even say its a set of any 50 50 bets not limited to coin tosses.

All of these are valid, the fact that we say this toss stands on its own or is part of a set has no bearing on the out come as our thoughts are not taken in account.. (or are they )


Single flip, or part of a set . not our choice.

To conclude, there are 2 possibilities, the toss is on its own or part of a sequence. the 2 contradict one another.
we cant just say it stands on its own because we feel like it. Also when a coin is tossed the results yield a 100% chance or being what ever it comes out to.

there is something missing here for randomness cant be predicted to 90 % and i dounght any one here would bet there 50% of there net worth on the third or 10th for that matter heads.

again i have not read all the posts maybe im missing something but trying to be open minded to both sides. i have no position on this at this point

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  #184 (permalink)
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traitor786 View Post
I understand what you are saying. Any flip has a 50 - 50 chance whether it is the first flip or the 10th. It is the laws of gravity airo dynamics and applied force. Nothing to do with math. Its random.

If the person who throws the coin does not know any specific turns and twists to influence it, it should be random.


traitor786 View Post
But then in a sequence things change. hhh is 0.125 suggesting that the flip after hhh has a 90% chance of being T.

No, not really. I am afraid you are confusing things. There are different probabilities:

-> the probability that a regular coin hits tails: 50%
-> the probability that the coin in the third toss hits tails; 50%
-> the probability (prior to the start of the first coin toss) that a sequence of 3 toin cosses will produce T-T-T: 12.5%


traitor786 View Post
But since it is a flip and all flips have a 50 50 chance we have a conflict.
Does the coin know if it is in a sequence ?

I have asked one of my coins and it was not aware that it had been in a sequence. Probably Alzheimer. Also there is no conflict, because the coin has no memory.


traitor786 View Post
If i take 3 coins and do 2 flips on each on and only one comes out HH then i take only that coin to my friends house and and bet him 80% of my money that it will be T will it be a high probability gamble ? or am i still at 50 50

You are still at 50/50. The coin has no memory.


traitor786 View Post
if i do 1000 coin tosses and end up with a 40-60 heads to tails situation what will happen when i change the coin. will the trend continue ?

A 40-60 heads to tail situation would be an extremely rare event. Let us do some calculations for fun:

10 coin tosses: There are 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1024 possible outcomes. Out of these 1024 possible outcomes there are 10! /( 6! x 4!) = 210 outcomes which have 4 heads. Accordingly you will find 120 outcomes for 3 heads, 45 outcomes for 2 heads, 10 outcomes for 1 head and 1 outcome for zero heads. If I add this up, I will get a total of 386 outcomes out of 1024, which stand for 40% heads or less. The probability can be therefore given as

P( 40% or less heads out of 10 coin tosses) = 386 / 1024 = 37.7 %

The number of outcomes that I used above can be directly taken from the last row of Pascal's triangle, as shown below:

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For 1000 trials it ain't no fun of calculating that same probability, but lucky enough I found a binomial distribution calculator, which is located here

Binomial Distribution: Probability Calculator

The cumulative probability that there are 400 heads or less is 1.3642320780 x E -10, or otherwise put it is below 0.000000013 %.

Did you imagine that you talked about an event, which will probably never happen in your lifetime? Screenshot of Binomial Distribution Calculator below.

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I am not responding to the remainder of your posts, as unfortunately you were confusing a lot of things.

I would suggest to have a look at some basic notions of probability calculus and statistics.


Last edited by Fat Tails; January 16th, 2012 at 02:48 PM.
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  #185 (permalink)
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Fat Tails View Post
If the person who throws the coin does not know any specific turns and twists to influence it, it should be random.


No, not really. I am afraid you are confusing things. There are different probabilities:

-> the probability that a regular coin hits tails: 50%
-> the probability that the coin in the third toss hits tails; 50%
-> the probability (prior to the start of the first coin toss) that a sequence of 3 toin cosses will produce T-T-T: 12.5%


I have asked one of my coins and it was not aware that it had been in a sequence. Probably Alzheimer. Also there is no conflict, because the coin has no memory.


You are still at 50/50. The coin has no memory.

.

Sorry for my confusion. I have limited knowledge in this and am just looking at logic putting aside all else for now.

I understand this can be frustrating for some one who understands the math. But in order to keep an open mind, I like to figure out the logic before I decide what formulae to use or theory to follow.

There is a flaw in this. It may not be mathematical, maybe its wording. Or maybe the incorrect wording may lead us to the wrong theory/formulae. Theories are made to be proven incorrect.

You state

-> the probability that a regular coin hits tails: 50%
-> the probability that the coin in the third toss hits tails; 50%
-> the probability (prior to the start of the first coin toss) that a sequence of 3 toin cosses will produce T-T-T: 12.5%

Is it just me or is this a contradiction?

odds that a coin lands TTT is 12.5 % Or .5 x .5 x .5 = 12.5%

I understand that this is the formulae but we must question why are we multiplying?

you state that the probability that the 3rd coin lands T is 50% That means that you are not taking in to account the sequence.

But then you say that prior to the start of the first coin toss the odds are 12.5 %. So here we are looking at a sequence.

If the odds are 12.5% that this can occur that would give a greater possibility that it wont occur. As soon as you say odds of a sequence are less the 50 50 you are saying that there is an imbalance.

If you trade and 60% of your trades are successful, then you would assume that things will revert to 60%

Reverting implies that there is some memory of some sort. The coin can revert to 50 50 unless it knows how much it went off by in one direction.

If indeed it does revert, then once it is at an imbalance say 40 T-60 H odds are that there will be more T's popping up in order to revert back to the 50 50.

You say that the odds are 12.5% of a sequence coming out TTT BEFORE the flips. Why only before?

Will the coin change its mind during the sequence ? how does our intention to start calculating before or after effect the toss. Maybe the coin can read our mind ?

I'm just teasing cause of the remark you made about my coins with good memory.

On the other hand,

It should be noted that the odds of T-H-T is also 12.5 %

and so is T-T-H

so when there is T T
odds the next flip will be H or T is EQUAL or 50 50 as you state

but, the issue is odds that the coin will land on either heads or tails is 100%

how ever 12.5% plus 12.5% is 25% not 100%.

Maybe the issue is my (or our) understanding of odds. maybe the formulae of .5 to the power of 3 is incorrect.

But

The fact that the coin will revert back to 50 50 after many flips is spooky to me. This is not my definition of random. Randomness should not revert by definition.

Maybe coins do remember. Maybe that's why strip clubs do so well. cause the money is happy, and the money conditioned us to keep going.

I'm really not expecting any replies any more lol Maybe I'm just teaching my self calculus the hard way here. But I will do a test with a real coin.

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  #186 (permalink)
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traitor786 View Post
You state

-> the probability that a regular coin hits tails: 50%
-> the probability that the coin in the third toss hits tails; 50%
-> the probability (prior to the start of the first coin toss) that a sequence of 3 toin cosses will produce T-T-T: 12.5%

Is it just me or is this a contradiction?

odds that a coin lands TTT is 12.5 % Or .5 x .5 x .5 = 12.5%

I understand that this is the formulae but we must question why are we multiplying?

We are multiplying, because the three toin cosses are independent of each other. Each toin coss has a probability of 50% of producing heads or tails. So any string - whether it is TTT or THT or HHH or HHT - has a probability of 12.5%.


traitor786 View Post
you state that the probability that the 3rd coin lands T is 50% That means that you are not taking in to account the sequence.

No I am looking at the single coin toss.


traitor786 View Post
But then you say that prior to the start of the first coin toss the odds are 12.5 %. So here we are looking at a sequence.

Correct, here I am looking at the sequence.


traitor786 View Post
If the odds are 12.5% that this can occur that would give a greater possibility that it wont occur. As soon as you say odds of a sequence are less the 50 50 you are saying that there is an imbalance.

Why? The odds of a sequence are not identical with the odds for a single toss. There is no imbalance.


traitor786 View Post
If you trade and 60% of your trades are successful, then you would assume that things will revert to 60%

Reverting implies that there is some memory of some sort. The coin can revert to 50 50 unless it knows how much it went off by in one direction.

If indeed it does revert, then once it is at an imbalance say 40 T-60 H odds are that there will be more T's popping up in order to revert back to the 50 50.

There is no reversion. Coins do not have a memory. If you toss a coin 1000 times and then 1000 times again, the second thousand tosses are independent from the result of the first 1000 tosses. Let us assume that the first 1000 tosses only produce 40% heads (very, very unlikely as shown in my prior post)

Then the next 1000 tosses are not affected by this, and will likely produce something around 50% of heads. The "reversion" is caused by the fact that after 2,000 tosses you now have 45% heads, 40% from the first series and 50% from the second series. But the second series was not affected by the first one.


traitor786 View Post
You say that the odds are 12.5% of a sequence coming out TTT BEFORE the flips. Why only before?

Before you do not know what will happen.

Probability for TTT before : 12.5%

Probability for TTT after the first coin toss with result heads: You now know that the 1st toss is heads. This reduces the probability for TTT to 0%.

Probability for TTT after the first coin toss with result tails: You now know that the 1st toss is tails. This increases the probability for TTT to 25%, as it now only depends on the second and third toss.


traitor786 View Post
It should be noted that the odds of T-H-T is also 12.5 %

and so is T-T-H

so when there is T T
odds the next flip will be H or T is EQUAL or 50 50 as you state

Correct.


traitor786 View Post
but, the issue is odds that the coin will land on either heads or tails is 100%

how ever 12.5% plus 12.5% is 25% not 100%.

Correct. The probability that the first two coin tosses produce TT is 25%.


traitor786 View Post
But the fact that the coin will revert back to 50 50 after many flips is spooky to me. This is not my definition of random. Randomness should not revert by definition.

As I said it is not reverting as this would be indeed spooky. The "reversion" does not affect the first series of tosses, it is just produced by the impact of averaging the first series with the second series.

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  #187 (permalink)
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I don't mean to offend anyone, but I would recommend this book as a starting point for understanding the basic concepts of statistics and probability.

http://www.amazon.com/gp/product/0062731025/ref=as_li_ss_tl?ie=UTF8&tag=mark01f-...&creativeASIN=0062731025

Don't be fooled by the fact that it's a cartoon; I read it (too many) years ago, and it really is a good introduction. It's also quite funny.

The author, Larry Gonick (http://www.larrygonick.com/), has done books on a lot of other topics too: http://www.amazon.com/gp/entity/Larry-Gonick/B000AQ75IY/?ie=UTF8&tag=mark01f-20&linkCode=ur2&camp=1789&creative=390957

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  #188 (permalink)
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Diiferent than the casino

Hi guys,
First post...here you go.
My colleague, an Ivy League PhD. in mathematics, and I have built a model based upon these concepts (50/50 coin flip).
We appied our algorithms and intraday trading rules to a global liquid futures market stream of tick data (325,000 actual market prices). We never took a position home overnight and were indifferent about market direction from day to day or week to week.
What we found is that quite different from the casino wager in which player risks $1 for $1 gain (yes in some cases you can buy odds, double down, lay chips on double 4s, etc...different topic), in our "Game" we always win a larger quanitity than the market (casino) wins from us.
Having searched for micro patterns of behavioral tendencies we arrived at a very specific set of rules (algo sets if you will) which put the odds of making money in our favor. Not the odds of winning %, but of making money.
What we found was that the more risk we were willing to endure, in order to increase our win %, the larger the drawdowns (negative strings I guess you would say) were and the more randomness we were exposed to.
The most interesting set of results was that over an 18 month, 8000 trade analysis using over 325,000 price points from a market data source, with a 40.7% win percentage using an intial capital stake of $25,000.00, our "Game" made over $228,000.00 with a maximum negative string of just under $13,500.00 and not a single negative month. We did not model progressive betting/trading in order to capture these returns. Always the original bet.
Summary: Since we, the players, can tell the casino (market) how much they will pay us if we are right and how much we will pay them if we are wrong, we found an optimal algo set which won less than 50% of the time and yet 'beat' the casino (market) by over 16:1 margin. Seems to good to be true, I know, my cynical 68 yr old mentor already told me....until he saw the reports...anyway, too long winded already. We are thinking about how to proceed...meaning keep this automated system to ourselves or open it up to others to participate with us. No we can't share the code, sorry, way too much invested in that...but if you have any interest we may put together a list of contacts of folks who like us are interested in a solid automated trading system.
And yes I do have quite a few day trading and chart setup ideas which I am happy to share...just not the code.

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  #189 (permalink)
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AlgoGeek View Post
Hi guys,
First post...here you go.
My colleague, an Ivy League PhD. in mathematics, and I have built a model based upon these concepts (50/50 coin flip).
We appied our algorithms and intraday trading rules to a global liquid futures market stream of tick data (325,000 actual market prices). We never took a position home overnight and were indifferent about market direction from day to day or week to week.
What we found is that quite different from the casino wager in which player risks $1 for $1 gain (yes in some cases you can buy odds, double down, lay chips on double 4s, etc...different topic), in our "Game" we always win a larger quanitity than the market (casino) wins from us.

Most of the casino games such as roulette are random games. As there is no strategy to win a random game with a negative expectation it is even useless to try. Black Jack is an exception, because due to the limited number of cards in a stack, the game is no more random when most of the cards have been played, but the expectation may shift into the positive territory.


AlgoGeek View Post
Having searched for micro patterns of behavioral tendencies we arrived at a very specific set of rules (algo sets if you will) which put the odds of making money in our favor. Not the odds of winning %, but of making money.
What we found was that the more risk we were willing to endure, in order to increase our win %, the larger the drawdowns (negative strings I guess you would say) were and the more randomness we were exposed to.
The most interesting set of results was that over an 18 month, 8000 trade analysis using over 325,000 price points from a market data source, with a 40.7% win percentage using an intial capital stake of $25,000.00, our "Game" made over $228,000.00 with a maximum negative string of just under $13,500.00 and not a single negative month. We did not model progressive betting/trading in order to capture these returns. Always the original bet.

Yes, I believe that there are micro pattern that you can observe and exploit. The question is only, if the population of traders and the species of algorithms will remain the same over the next 18 months. How will you adapt your rules, if they stop working because you have to share your meal with other carnivores?

Larger drawdowns are to be avoided because of their negative impact on progressive betting models. If you have a moderate edge, but relatively small drawdowns, you can easily leverage your returns by using an approach to bet half-Kelly.

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  #190 (permalink)
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So you cannot stack the odds in your favor by predicting strings. If that was the case, then it would be possible to 'stack' a future coin flip in your favor by purposefully creating a string beforehand that is biased towards your choice

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