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From: csc@watmath.UUCP (Computer Sci Club)
Newsgroups: net.physics
Subject: Re: Coin flips
Message-ID: <7784@watmath.UUCP>
Date: Thu, 17-May-84 12:26:04 EDT
Article-I.D.: watmath.7784
Posted: Thu May 17 12:26:04 1984
Date-Received: Fri, 18-May-84 00:15:59 EDT
References: <328@kpnoa.UUCP>
Organization: U of Waterloo, Ontario
Lines: 40


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 Hughes