<Language> Counting, Probability Theory, and the Birthday Paradox, #1

[H. Mark Hubey]

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I have this feeling that we have to re-do something we
started, partly for the benefit of the newcomers, and
partly for the benefit of the old-timers (no insult
intended). This re-enforcements is a necessary part of
learning. Social scientists often get angry that not
everything can be read like a novel.

Probability theory is basically fine-tuned commonsense
instead of raw commonsense. What I mean by that is that
the basics of probability theory follow from common sense
but the untrained mind extrapolates incorrectly. If this
happened in logic it is called a 'paradox'. Such names
do not exist in probability theory, (from now on PT or something
like it.) PT is normally taught starting with set theory
not the way it was developed. It is usually due to the fact
that most students taking such courses are mathematicians
and they find it easier (after years of training) to just
get the axioms and then build up from there. We started here
using intuition. The part of PT that many other people (especially
social scientists) are familiar with and also use, is called
statistics. Well, a "statistic" is really a number derivable
from a population of things. For example, the mean of the
population is a "statistic". The sample mean (the mean of a
random sample taken from a population) is also a statistic.

Statistics is obviously based on PT, because that is the underlying
basis of it. But it is often better to start off with some examples.

The basics are this:

1. probability it defined in at least two ways.
	1.a) frequency definition
	1.b) hokus-pokus definition
	1.c) mathematical definition

2. then we use these probabilities by combining them in various
ways. There are really only two ways
	2.a) addition or OR
	2.b) multiplication or AND

1.a) this is a limiting definition of sorts. For example, we imagine
an experiment in which a "fair" coin is N times. Then,
the probability of heads, written as P(H) is the fraction of the total
that the coin comes up heads as this number N approaches infinity.
First, nobody can throw a coin an infinite number of times. So we have
to only imagine it. Secondly, as we toss the coin many times and keep
a running ratio of H/N (where H=number of heads, N=total throws)
we will see that as N increases this ratio gets closer and closer to
0.5 as we expect.

1.b) That is where the hokus-pokus definition comes in. In such
problems we simply are either "given" P(H) or we make a determination
outside of probability theory based on symmetry arguments etc.

2.c) the mathematical definition formalizes this concept. We say that
there is a thing called a sample space. That is the space of all
possibilities. Obviously this is a set. And every element of the set
has some number associated with it, say wi (that's w subscript i)
such that when you add up all the wi, you get 1. This is a normalization
condition. That is because we consider probability to be a fraction.
If P(E)=0 then this event is the "impossible event". If P(E)=1
then this event is the "sure" or "certain" event. Therefore, if we
divide up the space of possibilities of an experiment into N sets
and then add them all up, we must get 1. That is because we are saying
that one of those things must happen, certainly. In the case of Heads
and Tails, (disregarding the possibility of the coin standing on its
side) there are two choices, H or T. Therefore we must have

1)	P(H) + P(T)=1

One of them must happen; that is certain. The readers must have
noticed that I sneaked in something here into eq.(1). That something
is that I did not say what '+' is. IT is in section 2, and the '+'
corresponds to what is meant by "logical OR'. So eq (1) means that
the coin will come up heads OR tails.

What is clear and certain here is that we have to have some way
of knowing that P(H)*P(T) is not 1/4 but zero. A coin cannot come up
showing both tails and heads, but applying the multiplication rule
above blindly would/could lead us to believe that. Some of what we
do has to come from "commonsense". However we can fix this like this.
Often unsolvable (or seemingly unsolvable) problems become easier to
solve and to understand if we generalize them. Suppose we generalize
this coin to an unfair coin. Then the probability of heads could be
some number x, between 0 and 1 i.e. 0< =x <=1. Then the probability
that it is not heads is 1-x. Therefore probability of both heads
and tails could be (note "could") x*(1-x). This is still wrong, but
it gave us fuzzy logic. In some cases, things are not so clear cut
and we want results like above. In our case, we have to think of the
sample space as being divided into two mutually exclusive sets so
that they both can't happen.

So here are two problems intimately related to many things being
discussed right now in historical linguistics. These problems can't
be solved so simply, but what is need to solve them is not beyond
high school math. We did this before, and we can muddle through
it again. It's better the second time around, they say :-)


Here they are: they are related to each other.

1. A marksman takes shots at a target so far away that we are
not sure that he will hit a bullseye. Suppose his probability of
hitting the bullseye is p. (Now, p is just a number here, a number
greater than or equal to 0 and less than or equal to 1. We normally
would write this as 0 <= p <= 1. In books the < sign and the =
sign are put on top of each other so that it doesn't look like
we have a representation of an arrow in <=)

a) What is the number of shots that he will [probably have to] take
before he hits the target (bullyseye)?
b) What is the probability that out of 10 shots he will hit 6?


2. The probabilities of birth of a male (boy) or a female (girl)
are approximately equal. Let's say that they are and that the
probability of the birth of a boy and the birth of a girl are
both 1/2 each. A couple have 5 children.

a) What is the probability 2 boys and 3 girls?
b) What is the probability of at least one boy?
c) What is the probability of getting 3 girls and then a boy?

(Problem 2.a does not ask for the probability of 2 boys being
born first, and then 3 girls or vice versa. The order does not
matter. Any combination of 2 boys and 3 girls will do.)


For those who are bored or tired, this is pretty much near
what you need to be able to do what is necessary to do some of
the simple calculations you see in articles like Cowan, Bender,
or Rosenfelder's article. A little bit more will allow you to
read Ringe's book. If you want to read Embleton's book, you have
to switch to statistics (which we can touch upon).




--
Best Regards,
Mark
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hubeyh at montclair.edu =-=-=-= http://www.csam.montclair.edu/~hubey
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