Warning – this post will be long, and probably quite
pedantic. Also, I assume most of you can google, so I won’t be posting links to
anything, as I feel you should look for it on your own, perhaps finding other
things that you find instructive or interesting on the topic. And, this is not
my best writing. I chose clarity over elegance (okay, I was just being lazy).
I had intended this post to be in one piece, but I don’t
have time to finish it today, and I had promised to get it posted. So I’m going
to slap a Part 1 on it and come back to it in a day or two. Sorry to those of you
who have been on pins and needles awaiting this exposition.
Background on Me:
I have both Bachelors’ and Master’s degrees in Mathematics,
and am working on a PhD in Mathematics Education. I have only recently begun to
identify myself as a mathematician, but I don’t consider myself a very good
one. But that is because I compare myself to others who are much more knowledgeable
and capable than I. In relation to the general population, I know a great deal
more about mathematics than at least 90%, probably more like 95%. I’m not
saying this to brag, but it is important to note, because the way non-mathematicians
understand and use mathematics can be quite different from those who have had
more training and experience in the subject. I also teach mathematics, currently at the
introductory college level, but I did teach high school for one year (and then
ran like hell). So, a lot of my assumptions about the general level of
mathematical knowledge is based on what I see in my classroom, which may not
reflect the composition of the overall population, or the gamer subpopulation in particular.
Background on Mathematics:
There seems to be a large misconception about what mathematics
is, and how it is applied. I don’t offer quotes much, but this is one of my
favorites “As far as the laws of mathematics refer to reality, they are not
certain; as far as they are certain, they do not refer to reality.” (Albert
Einstein) Mathematics is an axiomatic system – which means there are a basic
set of assumptions (axioms) upon which the body of theorems are built. There
are also definitions, which are used in building theory, and these definitions
must be precisely stated and applied. Many people have the belief that
mathematics is always black and white, right or wrong. This is not the case.
There are many areas of mathematics where disagreement can occur, often
stemming from the basic axioms and definitions. What is true, and this is the
focus of much higher order mathematics, is that once a theorem is proven
correctly, it must be universally accepted, but with the caveat that it is only
as good as the axioms and definitions upon which it rests.
Background on Statistics:
It is not unanimously accepted that statistics is a subgenre
of mathematics. I have read several articles discussing this issue, and my
opinion on this is not definitive, but statistics is as much about
interpretation as calculation. Stealing from a statistics professor I know, statistics
is the study of variation. There are a number of statistical terms that are
also words in the general lexicon, but the statistical meaning can be quite
different than the commonly understood definition. My point here is, people use
the terms of statistics in a manner that is not always correct or, more
importantly, precise. Precision of language is of the utmost importance in
mathematics!
Background on this Post:
I have noticed on forums, that probability theory is not
always correctly applied or calculated. Furthermore, there seems to be a lack
of understanding of the different kinds of probability and a misuse of
statistical terms. I’m not calling anybody dumb, but it inspired me to write
this post as a means of describing some of the ideas behind probability theory and
explain some of the mathematics involved in making calculations. I will also
offer my opinion on the feasibility and usefulness of applying probability
within the context of gaming. I’m assuming a level of mathematical knowledge
that has Algebra as a background, but I will try to be as clear as possible for
people that haven’t seen this before.
Famous Problems in Probability
Perhaps the most well-known probability problem is the Monty
Hall problem. This is based on the television show “Let’s Make a Deal”, where a
contestant is given the choice of three doors.
Behind one door is a prize, behind the other two is sometimes a goat (I
haven’t seen the show in many, many years). I will refer to the prize as good
and the other two as bad. So, given three options, the contestant chooses one.
The host will subsequently reveal one of the remaining two doors, which is
always going to be bad, as in no prize.
This is important. There is a 1/3 chance that the contestant has the
prize, while a 2/3 chance that the host does. There is a 100% certainty that
the host has at least one door with no prize. When looking at a deck of cards,
probability can be reassessed based on information that is revealed. In the
Monty Hall problem, no information is revealed. The host will always have a
door with no prize to reveal. After the reveal, the contestant is given the
option of trading their door with the one the host still has not revealed.
Probability is in favor of the host holding the prize, so according to the
probabilities, the contestant should always trade. This is a contentious problem, and some very
well-educated people still can’t agree with this conclusion. It took me a while
to come to terms with it. It is natural to say that there is an equal chance
that the contestant or host holds the prize, 1/2. This is not however the case,
the initial probabilities still apply. As I stated, the important thing is that
no information is revealed in this case. Information will be key to later
arguments.
For those not convinced of the truth behind the Monty Hall
problem, I offer an alternate scenario. Say there are 20 doors. The contestant
chooses one, so they have a 5% (1/20) chance of holding the prize, while the
host has 95% (19/20). If the host reveals 18 doors with no prize behind them,
would you still feel the contestant has a 50% chance of winning?
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