sharp@kpnoa.UUCP (05/14/84)
<>
Ethan's comment last month on the rational way to approach a series of coin
tosses that went T,H,T,H,H,H,H, and that he'd choose heads because the coin
might be biassed, brings to mind a whole branch of probability that I've
never had time to investigate. Perhaps someone out there knows ....
The question is this: what is the probability that the coin IS biassed ?
In general, given a sequence with a predicted a priori probability of any
particular event (for a coin, prob=0.5 of either head or tail), what can we
say about this hypothesised probability ? Is there a statistical or
probabilistic test which gives us a level of confidence in our assumed
probability given a realisation of the sequence ?
After all, I remember being in a fairground with a machine which was stuck,
and always came up on one of three positions out of more than a dozen. It did
not take us long to empty the machine, since realising the problem was easy !!
While I'm soliciting information from the net, comments about the book
"The Fifth Generation" and its remarks about nets will be welcome.
Nigel Sharp (noao!sharp, for a little while yet kpno!sharp)csc@watmath.UUCP (Computer Sci Club) (05/17/84)
The short answer to Nigel Sharp's question, is there a procedure by
which, given the results of flipping a certain coin a number of times
(say H,T,H,T,H,H,H,H), we can place "confidence" limits on whether the
coin is fair or not, is yes. To be much more specific would require
the equivalent of a first course in statistics (the question asked is
the fundemental question of statistics, given observed values of a
random variable, what can we say about the distribution of this
variable).
I will try and outline the procedure. In this case all we are
concerned with is the number of tosses, and the ratio of heads to
tails obtained, (in the above example 8 and 6/2 respectively).
Assume the probability of heads with the coin is p. Calculate the
probability of getting six heads and two tails, GIVEN THAT THE
PROBABILITY OF GETTING HEADS IS p. Call this value l(p) the
"likelihood of the coin having a probability of p of giving heads".
This value does not mean much by itself. Do the same calculation
for all p between zero and one. Take the resulting function l(p)
and find the point s such that l(s) is the maximum. That is the
value of p such that the probability of the observed outcome is
the greatest. This is called the maximum likelihood estimator.
In this case it is easy to show (example numero uno in any statistics
course) that s = (number of heads)/(number of tosses). This is
usually (but not always) considered the best estimate of
p.
In most cases there will be many values of p for which l(p)
is close to l(s) (that is many "reasonable" values for p). Therefore
rescale l(p) such that its maximum value is 1 (in general the maximum
value will be much less than 1). Call this new function L(p) the
relative likelihood. If L(q) is 1/2 it means that the l(q)/l(s) is
1/2, that is the value q is "one half as likely" as the value s.
Conventionaly we dismiss any values p with L(p) < 1/10, that is
values that are less than on tenth as likely as the most likely
value. Thus if L(1/2) were less than 1/10, we would conclude that
the coin was biased. If L(1/2) were greater than 1/10, we would
conclude that "we do not reject the null hypothesis that the coin
is unbiased".
Isn't statistics wonderful.
William Hughesdon@allegra.UUCP (D. Mitchell) (05/18/84)
The question about coin flips is a good one. There are two major
answers depending on which to the two philosophical schools of
statistics you belong to.
Classical statistics has something called "hypothesis testing". You
know a fair coin yields a binomial distribution which will look like a
bell shaped curve (symmetric when Prob(heads) = 0.5). You pick a
confidence level, say 99 percent, and then you reject the fairness
hypothesis if the result of your coin flips is out in the 1 percent
tails of the curve.
The flaw in that approach is that picking the 1 percent rejection area
is arbitrary. What if the distribution is uniform? (not for a coin
flip, but for some"experiment"). The concept of confidence breaks
down.
Bayesian statistics deals with the problem more consistently (in my
opinion), but has bizarre philosophical implications. A Bayesian
believes that human knowledge is described by probability
distributions. That is, probability is subjective, how strongly you
believe something will happen.
When someone gives you a coin, you think it is fair, so your own
private "prior" distribution is binomial. When you do an experiment,
you can take the results and use them to mathematically transform the
distribution into a new one ("the posterior distribution"). If the
coin gives 40 heads out of 40 flips, this distribution will be strongly
skewed.jim@ism780.UUCP (05/21/84)
#R:allegra:-248100:ism780:20300002:000:878 ism780!jim May 19 21:49:00 1984 > Bayesian statistics deals with the problem more consistently (in my > opinion), but has bizarre philosophical implications. A Bayesian > believes that human knowledge is described by probability > distributions. That is, probability is subjective, how strongly you > believe something will happen. I don' find the implications bizarre at all. It seems clear to me that probability is a measure of lack of knowledge; the probability that heads shows on coin which I have flipped but am covering is very different than the probability once I reveal it. And if I have flipped it a million times beforehand to determine its bias, or have loaded it somehow, then the probability *for me* is different than what it is for someone else, and for us to give the same answer to the question "what is P(heads)" would be bizarre indeed. -- Jim Balter, INTERACTIVE Systems (ima!jim)
steve@Brl-Bmd.ARPA (05/24/84)
From: Stephen Wolff <steve@Brl-Bmd.ARPA> Oh, just wonderful! The blurry physicists are going to fight the Bayes/classical battle all over again. Foo. 'Bye.
gwyn@Brl-Vld.ARPA (05/24/84)
From: Doug Gwyn (VLD/VMB) <gwyn@Brl-Vld.ARPA> The way I learned probability theory, probabilities are "conditional". P(a|b) denotes the probability that a is true, given that one knows that b is definitely true. The usual laws of probability then are P((not a)|b) = 1 - P(a|b) if "a or (not a)" is a tautology P((a or c)|b) = P(a|b) + P(c|b) - P((a and c)|b) P((a and c)|b) = P(a|c) * P(c|b) If all terms in a given context end in "|b" then the "|b" is often omitted, and assumed to be a condition in the environment. Baye's theorem follows trivially from (a and c) == (c and a): P(c|a) = P(a|c) * P(c|b) / P(a|b) And so forth. The idea of conditional probability is a formalization of the (obvious) fact that a probability assessment must be dependent on one's state of knowledge of relevant factors.