Independent Events-Probability Question

HI,


We all know that if you flip a coin ten times in a row, all heads that the probability of getting tails on the 11th toss is still 50%. This is because the flipping of a coin is an independent event (i.e the coin has no memory of what has happened in the past)


There is a counter argument that goes like this: If in the long run we expect the number of heads and number of tails to be equal (50/50) and after , for example 10 tosses of all heads, that in order to equal out there must be more tails expected than heads in the future. In other words, the chances of getting tails have increased after the 10th toss.


Now, I know this is a bogus argument, but why??? If I remember correctly there is a name given to this argument and I cannot remember where I read it.


Can anyone help ... thanks

To become really a 50/50 chance you have to do the experiment infinite times. And that's exactly where the problem of the counter argument lies IMO. It simply doesnt matter if you have done it 10 or 100 or 1000 times, because you have to do it infinite times to make it exactly 50/50 (infinite minus 10, 100 or 1000 is still infinite). When doing finite experiments your expected probability would be 0.50, but there is a fault margin (dont know if that is the english word for it, but I think you know what I mean) dependend on the sample size. This margin will always have a positive value greater than 0, except for an infinite sample size.


Regards

-- "If in the long run we expect the number of heads and number of tails to be equal (50/50) and after , for example 10 tosses of all heads, that in order to equal out there must be more tails expected than heads in the future. In other words, the chances of getting tails have increased after the 10th toss."


-- Allow me some elaboration. It's a Sunday..


First of all, statistical science does not PREDICT the future. It merely offers us the best possible GUESS as to what we can EXPECT to happen.


Second, we do NOT expect to have exactly 50/50 results after, say, 20 flips. On the contrary, statistical science suggests that this is an unlikely event. Which does NOT mean that it can't happen! (See first point.)


Third, we must decide from the start if we believe that the coin has a memory or not. If we DO believe that the coin has a memory *, then indeed the results must converge until they reach equality (i.e. exactly 50/50) after a number of flips. If we do NOT accept that the coin has a memory, then we must treat every flip as starting a new series of flips, each series completely indifferent to what went down before. That's your standard amnesiac coin.


Fourth : Although it is not correct, per statistical science, to claim that the results will be PRECISELY as expected (i.e. "50/50") in the long run, we CAN claim that the results will CONVERGE towards our mathematical (theoretical) expectation, or prediction. Which means that, as the flips go on and on, the results will converge towards 50/50 more and more.


Fifth : The above point does not mean that the absolute numbers themselves will converge but, rather, that the percentages will.


In other words, what statistical science says is this: We flip the coin 100 times. Let's say we get 64 heads and 36 tails. We flip the coin 900 times more and we have, overall, in 1000 flips, 580 heads and 420 tails. Notice that the difference in the numbers between heads and tails widened significantly, from 64-36=28 flips to 580-420=160 flips. But the difference in the percentages has decreased, from 64%-36%=28% to 58%-42%=16%. The percentages converged closer to 50%/50%.


Bottom line : According to accepted statistical theory, if you flip a honest coin 10 times and it comes up 10 times Heads, then the next flip has a chance of being Tails ..50% again. (And check that coin again, while you're at it!)

_____________________________________________


* A German philosopher, called K. Marbe, at the beginning of the last century wrote a book titled "The Uniformity Of The World". In it he argued that the cosmos does indeed have a "memory"! According to Marbe, after 10 consecutive flips of a honest coin coming up Heads, then Tails becomes more likely. IMPORTANT NOTE: Marbe's argument cannot be refuted by logic but is rejected because of lack of empirical support! argument cannot be refuted by logic!


The same goes for the argument in favor of the theoretical model of the Bernoulli scheme of trials (i.e. non dependence of outcomes, only two possible outcomes, constant probabilities throughout the trials) : it is also supported by experimental evidence rather than by logic.


K. Marbe's name is mentioned in Feller's book.

The existance of the casino I go to, and any casino for that matter, is good enough proof IMO that no human is able to predict the future.


If probabilities were changed by past outcomes, then the casino would long ago have been broke. Long long before you were born even.

--"The existance of the casino I go to, and any casino for that matter, is good enough proof IMO that no human is able to predict the future."


The existence of the casinos seems to prove that humans, or at least statistical science, CAN have a reliable estimate of what the future holds for them in games of chance. (Aside to Down Under: I would still not call this a "prediction", though!)


Statistical science is all a honest casino has going for it - plus a bankroll that dwarfs the players' bankrolls. That bankroll also decreases to a very safe level the casinos's Risk of Ruin from the fluctuations of the players' fortunes. So, yes, the casino HAS 'predicted' the future ; it cashes in its predictions every day.


--"If probabilities were changed by past outcomes, then the casino would long ago have been broke."


Absolutely.


Only, of course, you are referring to games where independent events take place, such as Roulette, played with a honest (unbiased) wheel. This is not the case in games like Blackjack, where future events are affected by past ones, and other beatable casino games.

the answer is that the results do not have to exactly even out, but only that they will converge to 50%. In order for this to happen you do NOT need to have more heads than tails.


If you flip a coin 100 times and get 100 heads, you do not need more tails than heads for the number to converge to 50% in the long run. If you have 1,000,000 more trials, and the split is exactly 50-50, then the total is 500,100 heads to 500,000 tails, which is as close to 50% as you can get. The numbers converge without more tails than heads occurring.


Pat

JM,


The posters above seem to have the concept nailed but I believe the name for it is "regression towards the mean". I think it is in Sklansky's "Getting the Best Of It" but I could be wrong.


Regards,


Rick

Consider this: If someone believes a tail is more likely after getting ten heads in a roll try this bet. You toss a fair coin until you get ten heads in a row, and then you bet the next toss is a head and your freind bets it is a tail. Since he believes it a favorite is willing to lay odds. See what happens...