From Barry Greenstein’s insightful poker book, Ace on the River:
Someone shows you a coin with a head and a tail on it. You watch him flip it ten times and all ten times it comes up heads. What is the probability that it will come up heads on the eleventh flip?
A novice gambler would tell you, “Tails is more likely than heads, since things have to even out and tails is due to come up.”
A math student would tell you, “We can’t predict the future from the past. The odds are still even.”
A professional gambler would say, “There must be something wrong with the coin or the way it is being flipped. I wouldn’t bet with the guy flipping it, but I’d bet someone else that heads will come up again.”
Yes I know the math student would really say “individual trials are uncorrelated,” not “we can’t predict the future from the past.” The lesson still holds.
Happy Labor Day, everyone.
@jt512 23, one defines X and Y to be uncorrelated when the covariance Cov(X,Y)=E(XY)-E(X)E(Y)=0. Lets assign Heads=+1 and Tails=-1. Then in the case in which the probability of Heads is 1, then E(XY)=1 and E(X)=E(Y)=1, and therefore Cov(X,Y)=1-1*1=0, and so they are clearly uncorrelated. jt512 seems to be deeply confused about basic statistics; this would be fine, if not for the fact that jt512 is putting themself out there as some kind of expert and so critical of everyone else.
Well, Fred, I guess I have to concede that you’re right here. Most references that I can find that define “uncorrelated” defines it in terms of the covariance, rather than the correlation coefficient.
I don’t think, though, that that misunderstanding implies that I’m “deeply confused about basic statistics.”
ok jt512, sorry if i seemed impolite.
This is a really important point.
In all kinds of thought experiments and logic puzzles (and in, like, every paper ever in moral philosophy), we’re told, “Assume you’re in situation X” and then asked to reason in some way from there. But the real world is never like that. Whatever information we have, we’re using not just to draw some conclusion conditional on being in situation X, but also to update our prior as to whether we really are in X.
What makes it trickier is that in social contexts we all have some very strong priors that we can’t articulate clearly, or that we’re not even conscious of.
In a strictly frequentist approach, you would say that based on the data, the maximum likelihood estimate of the probability of the coin coming up head is 1, so there is no reason whatsoever – purely based on the data – to bet on tails.
In a Bayesian approach, you would look at the probability of the coin coming up head as a random variable, with is own prior distribution. The prior can be anything that describes your prior assumptions about the coin, before observing any flips. Some people may be very confident that the coin must be fair and they trust the coin flipper to be fair too – this would mean that their posterior belief, after observing 10 consecutive heads, would only shift by a little bit towards heads. Other priors are also possible.
However, one thing is clear: unless your prior distribution has a bias towards TAILS, you will always infer that a HEAD is more likely on the 11th flip.
I agree with Ben.
You can take two views:
a. to let yourself be guided by empirical data: in this case, bet on heads.
b. to let yourself be guided by your theoretical understanding: I have no clear evidence to suggest that the coin is not fair, so equal probability. If I have, I will act according to a.
This hinges on what “no clear evidence is”. If you define it as no falsifying evidence, heads coud come up a millions time and you don’t change your stance, because it is not impossible for this to happen with a fair coin. But if a close analysis of the physical coin suggest a physical difference, this will falsify the hypothesis “the coin is not fair” and you act according to a.
Probability is an integral part of a 4D cross-section of the universe- right down to quantum mechanics! The fact that nothing is completely certain or absolutely impossible, however offers a clue to the possibility that what happens in reality is just about completely certain. The chance that the sun will rise tomorrow appoaches 1, yet there was a time when the sun did not rise and there will be, in the future, a time when the sun no longer rises. My existence was next to impossible-almost 0- but here I am!
Each time a coin flips, so long as great pains are taken to keep the flipping process random, the chance of heads vs tails is exactly the same. However I remind my students a coin has three sides, not two and the chance the coin will stand on end increases as the thickness of the coin increases. If a coin is thick enough and becomes a cylinder, landing on the ends becomes very improbable- and landing on its third side becomes almost certain.
Seen from its proper perspective….its complete perspective, the universe much more clsely approximates a cylinder of vast length than a coin….
A few additional thoughts….
1. The chance that a cylinder of vast length would stand on either end approaches zero
2 The chance that the cylinder will remain on its third surface approaches one.
3. The two ends are plane surfaces
4. The third surface is circular
5. The lower dimensional similarity to Euclidean SRT….local flat space and GR with its marginally closed spheroid geometry.
6. The overall stability of the sytem
7. The possbility of motion and change as the cylinder rolls in a certain direction…..
“A professional gambler would say, “There must be something wrong with the coin or the way it is being flipped. I wouldn’t bet with the guy flipping it, but I’d bet someone else that heads will come up again.””
Jimmy the Greek said that he could make money from gambling in Vegas because he never played the games; he would do things like “I bet that blackjack player will bust on the next round” and make bets on the outcomes of the games.
It depends on what your prior probability is for the hypothesis, ‘the coin tosses are all fair’ (depending on whatever other information you would use if this were a real world situation). If, for whatever reason, you take that prior probability very very close to 1 (say because you’d tested the coin yourself), then you’d predict the heads-on-next-flip probability to be very close to 1/2. The professional gambler is a cynic and assigns a nonzero probability to unfairness somewhere. She has the right strategy provided it’s an even money bet. She would be a fool to pay odds to someone though, unless she had evidence of unfairness. Put it this way, the odds of 10 heads in a row is 1024:1, but far fewer than 1 coin in 1000 is unfair. So a random coin which results in 10 heads in a row is still likely to be fair. (The most likely explanation for a random coin coming up heads 10 times is that the coin is fair and there was a statistical fluctuation.)
But if it is 100 heads in a row, Hamlet is having you killed.