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Probability Paradox!?!

  • #61
  • Jan 18, 2012 12:41am Jan 18, 2012 12:41am
i think the mistake comes from the fact that

50% is the constant event no matter what the past or what the future data, if the event is truly random then coin flip having 2 sides and only able to land on 1 side will always yield 50% (assuming the coin is tossed with the same force, the sides weight equally wind etc...all remains constant, and random)

we are able to calculate the probability of % of heads and tails on a 9 toss series however once we toss the coin 50% is our probability independently of our prior calculations. any streaks are pure luck and is not an indication of future events.

at the end of the day the math should cancel off leaving the probability of heads to be 50% as in tails so in 9 tosses we expect 4 tails 4 heads and 1 completely random toss, however we could have 9 heads in a row and that would be just luck as the actual coin toss is 50/50

2 sides divided by one possible outcome
AVT INVENIAM VIAM AVT FACIAM
  • #62
  • Jan 18, 2012 12:54am Jan 18, 2012 12:54am
let me put it in a simpler way....for those who think that streaks are possible and an edge can be attain from RANDOM data then answer this question

on a coin........there are 2 sides

can only fall on one side right?

so what is the force, that can affect this outcome in a way that streaks are possible and somehow making it more likely that one side is more probable than the other side, when 2/1 is 50%

in other words how can the past affect this equation? 2/1, what force makes 2.8/1 or 1.5/1 possible when 2 sides 1 outcome?
AVT INVENIAM VIAM AVT FACIAM
  • #63
  • Jan 18, 2012 3:02am Jan 18, 2012 3:02am
Quoting Custos
If maths isn't the purest of common sense, then I don't know what is. Go to a maths professor and ask what's the likelihood of tails coming up after a million tails in a row. it is ALWAYS 50%, no matter what happened before.
After a million consecutive tails, I might be tempted to assume that the coin is not completely fair. Perhaps that is what jam*3 meant by "We shouldn't get too bogged down in any theory to lose sight of common sense." Just pondering.
  • #64
  • Jan 18, 2012 3:18am Jan 18, 2012 3:18am
Quoting Roofx
What about this for an idea of 'edge':

You have a simple coin flip betting experiment. In theory there is a 50% chance of getting heads and a 50% chance of getting tails. If you guess correctly you win $10. If you guess wrong you lose $11 (consider the extra $1 as transaction costs). What will happen over time? You will lose your money.

Now, what if we factor in the concept of series' of flips as has been mentoned in the thread instead of taking each flip as a single event. What if we say the probability of getting heads on the next...
Market behavior and probabilities are not going to change merely because one retail trader has had a recent series of losses.

Hence you'd need to identify and prove a causal link between your losing and winning trades, i.e. demonstrate that a win has > 50% probability of occurring, following X consecutive losses, beyond the realms of 'normal' statistical fluctuation. Otherwise you run the risk of being 'fooled by randomness'.

Instead, if I was the trader, I would be looking to examine (and possibly overhaul) my strategy in an attempt to reduce the number of losses that occurred.
  • #65
  • Jan 18, 2012 1:09pm Jan 18, 2012 1:09pm
Quoting Custos
If it works as you suggest, then why don't you go to the casino and only bet on red after ten blacks in a row. I assure you, in the long-term you would still lose. Past events, in this case, are not linked to future events.
You're still missing his point. There comes a time when theory and practicality run into each other-- if you have a physical coin that flipped and landed on tails one million times in a row, there's a pretty damn good chance that the coin's probability of landing on either side is not actually 50/50, and that has to do with the physical coin itself.

And to add a real life example, documented gamblers have had extensive success going to casinos and observing specific roulette tables over time to see that specific numbers DID come up more often than simple probablity theory would suggest, and they would then hit those tables and bet the same numbers over and over. I believe it had to do with the fact that over time roulette tables wear down and certain "paths" are created that lead to certain numbers hitting at a higher occurance than what the maths would suggest.

Given that the market isn't some mathematically perfect machine, it would probably be wise to approach it in much the same way as the coin that flipped tails a million times in a row...
  • #66
  • Jan 18, 2012 1:13pm Jan 18, 2012 1:13pm
Quoting smittens4212
You're still missing his point. There comes a time when theory and practicality run into each other-- if you have a physical coin that flipped and landed on tails one million times in a row, there's a pretty damn good chance that the coin's probability of landing on either side is not actually 50/50, and that has to do with the physical coin itself.

And to add a real life example, documented gamblers have had extensive success going to casinos and observing specific roulette tables over time to see that specific numbers DID come up more...
again you failed to answer the question

a coin with 2 sides with only one possible outcome will forever be 50/50
2/1= 1/2=0.5=50%

now what FORCE except your imagination or belief tilts this equation any different?
AVT INVENIAM VIAM AVT FACIAM
  • #67
  • Jan 18, 2012 1:31pm Jan 18, 2012 1:31pm
Quoting smittens4212
Given that the market isn't some mathematically perfect machine, it would probably be wise to approach it in much the same way as the coin that flipped tails a million times in a row...
then please do so. Nobody is stopping you from attempting that.
  • #68
  • Jan 18, 2012 2:54pm Jan 18, 2012 2:54pm
Quoting the redlion
again you failed to answer the question

a coin with 2 sides with only one possible outcome will forever be 50/50
2/1= 1/2=0.5=50%

now what FORCE except your imagination or belief tilts this equation any different?
You're right if you know beforehand that this coin has a 50% chance of landing on either, but to be honest I skipped your posts on the matter and did not see that you specified knowing a 50% chance on the flip. Also, without getting into the physics behind it, you can not bias a coin on a flip, no matter how you alter the weight. The way I interpreted the example is we see a coin flip (a theoretical one of course) and it lands on tails one million times in a row, despite our belief that a coin flip should have a 50% chance of either. After seeing this (despite it being physically impossible, but bare with me), you'd want to reconsider your belief that this coin is non-biased, hence his point "At some point you need to correct your model to reflect reality". Substitute playing with unknowingly loaded die, and it's the same thing.

The roulette example still stands though, and serves to prove the point even better because it IS a real life example. Given the theoretical odds of roulette you have x chance of y happening, however after actually observing a roulette table (or tables), you'd be quite wrong to continue betting the odds in the face of evidence showing these odds likely aren't still reflected in reality.

Quoting Custos
then please do so. Nobody is stopping you from attempting that.
If you really did play an odds game where you believed the odds to be X, but after 1000000 consecutive examples it turns out your odds aren't actually X or even conceivably close to X, but you kept betting on the X odds, I believe that is the textbook definition of insanity-- doing the same thing over and over expecting a different result. Of course, nobody is stopping you from attempting that, either
  • #69
  • Jan 19, 2012 12:02am Jan 19, 2012 12:02am
Quoting smittens4212
If you really did play an odds game where you believed the odds to be X, but after 1000000 consecutive examples it turns out your odds aren't actually X or even conceivably close to X, but you kept betting on the X odds, I believe that is the textbook definition of insanity-- doing the same thing over and over expecting a different result. Of course, nobody is stopping you from attempting that, either
Yeah smittens, I was actually talking about a game with 50% chance. Then let's just flip a coin with a program, if you have problems with a physical coin getting flipped a couple of times.
Now, what you don't seem to get is, that also totally random events can have extensive streaks, but the irony is that just because the computer hit head a thousand times or more, it doesn't mean that the next time tails is going to appear. The chance is always 50%. And now don't come up with something that the computer is programmed wrong or blabla.
I am talking about totally random events. They can have streaks of course, but you can't imply the future outcome by such streaks. The chance IS ALWAYS 50% no matter what happened before. And yes a million tails or heads in a row, as unlikely as it is, can happen as a streak. It is mathematically not impossible. But the next coin toss will still only have a 50% chance of hitting tails.
That is also why a lot of systems that are actually producing random results, get very popular for a couple of months to years. Most people don't understand that they have a lucky streak that lasts over a couple of hundred to thousand trades and then the system goes back to random. In fact the system always produced random results, it just had a lucky streak within the random results.
  • #70
  • Jan 19, 2012 1:13am Jan 19, 2012 1:13am
Quoting Custos
Yeah smittens, I was actually talking about a game with 50% chance. Then let's just flip a coin with a program, if you have problems with a physical coin getting flipped a couple of times.
Now, what you don't seem to get is, that also totally random events can have extensive streaks, but the irony is that just because the computer hit head a thousand times or more, it doesn't mean that the next time tails is going to appear.
I think he gets it, but he's just talking about the real world. In the real world, there is a 100% probability that any coin is unfair.
The nail that sticks out gets hammered back in
  • #71
  • Jan 19, 2012 3:34am Jan 19, 2012 3:34am
In fact in the real world the coin is more predictable than most people assume.

Quote
But using high speed cameras and equations, Diaconis and colleagues have now found that even though humans are largely unpredictable coin flippers, there's still a bias built in: If a coin starts out heads, it ends up heads when caught more often than it does tails.
http://www.npr.org/templates/story/s...toryId=1697475
  • #72
  • Jan 19, 2012 12:46pm Jan 19, 2012 12:46pm
Quoting Custos
Yeah smittens, I was actually talking about a game with 50% chance. Then let's just flip a coin with a program, if you have problems with a physical coin getting flipped a couple of times.
Now, what you don't seem to get is, that also totally random events can have extensive streaks, but the irony is that just because the computer hit head a thousand times or more, it doesn't mean that the next time tails is going to appear. The chance is always 50%. And now don't come up with something that the computer is programmed wrong or blabla.
I am talking...
I get it, I think our misunderstanding comes from where we start-- my base assumption for this theoretical example is that we're given a coin, or die, or cards, or any other odds game, and we assume (but don't know) that the odds are fixed equally, i.e. heads in a coin toss is 1/2, six-sided die is 1/6, etc. However after repeated observation, our assumption that the odds are fixed equally may need to be revisited. If we know the odds are fixed at a level, they're fixed, no matter if it's 1000000 or 1000000000 consecutive sequences. If you rolled 1000000 sixes in a row on a die, though, maybe it's worth considering that the odds are fixed-- just not where you think they are.

Not particularly interesting, I'm bored at work, bare with me
  • #73
  • Edited 1:51pm Jan 19, 2012 1:32pm | Edited 1:51pm
Quoting smittens4212
I get it, I think our misunderstanding comes from where we start-- my base assumption for this theoretical example is that we're given a coin, or die, or cards, or any other odds game, and we assume (but don't know) that the odds are fixed equally, i.e. heads in a coin toss is 1/2, six-sided die is 1/6, etc. However after repeated observation, our assumption that the odds are fixed equally may need to be revisited. If we know the odds are fixed at a level, they're fixed, no matter if it's 1000000 or 1000000000 consecutive...
Smittens,
You are spot on in all of your replies. The issue I have with gambler's fallacy, is that once people finally get it, then they take it to the opposite extreme, without trying to understand the practicality of it. That's why I originally countered the one million flip streak comment. Glad to see someone actually gets it.

It's sort of like zeno's paradox. Would someone let an archer aim and release an arrow five feet away from their heart because some mathematician assured them that it would 'theoretically' never reach them? I hope not.

For the people who keep chiming in about the 50% assumption, yes, we get it. And hopefully you also agree with the pragmatic real world and bayesian revisions. Given a computer and infinite time, one could by chance hit a streak of 10 trillion heads in a row and the next toss would be 50/50. Yes that's perfectly true, but not well grounded in physical reality.

Btw. you probably already know it, but Ed Thorp and Claude Shannon actually used the roulette wheel imperfections to make money in casinos!
  • #74
  • Jan 21, 2012 2:27pm Jan 21, 2012 2:27pm
thinking in probabilities amazes me.but i think those monte carlo simulations take it too far.
Just try to imagine the fact that in 100 years we will all be dead,in fifty years over half will be waxed.given the average trading life is rarely a decade long,some of this scenarios will happen only if you live a milllenium.
The probability of losing money trading is so high thats its foolish for a sensible man to attempt.if 10% make it why try?
But we are all here,united by greed,regardless of fancy philosophies of why we trade.

Therefore it seems to me we all live on hope and downplay the odds.in other words the choices we make are not logical but based on possibility and not probabilities.
wo-yoy! wo-yoy! wo-yoy! wo-yoi! wo-yoy-yoy-yoy!
  • #75
  • Feb 16, 2014 9:38am Feb 16, 2014 9:38am
This thread is pretty interesting. I thought it'd be interesting to bring in some of Kahneman and Co's perspectives from their forays in psychology/behavioural economics.

The conception of this thread immediately reminded me of heuristic - shortcuts our brain takes instead of performing tedious and rigorous manual calculation.

An important and relevant bias caused by heuristics is the Law of Small Numbers and the Gambler's Fallacy

- Toss of a fair coin should yield roughly 50% heads - 50% tails following the law of large numbers
- With 10 tosses, we are wired to incorrectly think that the sequence of 10 tosses should be representative of the underlying Bernoulli distribution
- One would find a sequence of H,T,T,H,T,H,T,H,T,H more likely than T,T,T,T,T,H,H,T,H,T even though they are equally likely!
- This explains the incorrect belief at the roulette table of 'mean-reversion'

On the other side of the coin (excuse the pun) would be the fallacy of Hot Hands where we incorrectly form beliefs about trends on things which are actually merely random - e.g. lottery numbers

Lottery makes a pretty interesting example as one can fall into either categories of incorrect beliefs
  • #76
  • Last Post: Feb 16, 2014 3:01pm Feb 16, 2014 3:01pm
Quoting hanover
{quote} Reactor, You're right about the lottery, but Agro's comments about roulette in post #35 are correct. With one 0 on the wheel, the house odds are always 2.7% against the player, regardless of whether he bets on red/black, a whole row, a whole column, 4 numbers in a square, a single number, etc etc. Here's what you're missing: Odds are calculated from expectancy, not probability. If a player bets on a single number, he will be correct 1 time in 37, and wrong 36 times. Hence you're correct about the probability being 1 in 37, or 3%. But let's...
Thanks for this post hanover. It explains why in a TE you can have have a very high win rate and a negative equity curve.
If you are good at something, never do it for free--Joker
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