Chapter Seven Notes
Discrete Probability
Based on: Discrete Math & Its Applications - Kenneth Rosen
CSC125 - Spring 2012
7.1 An Introduction to Discrete Probability
Basic definition – p. 446
P(E) = |E|/|S|, where S is the set of all outcomes
Theorem 1 – p. 449
P(not-E) = 1 – p(E)
Theorem 2 – p. 449
P(E1  E2) = P(E1) + P(E2) – P(E1  E2)
Monty Hall 3 door Puzzle – p. 450
Let R be you chose the correct door;
W be you chose the wrong door.
P(R) = 1/3
P(W) = 2/3
If you do not switch, your probability of winning is 1/3
Now, suppose you switch:
If you chose correctly, you lose  with p = 1/3
If you chose wrong, you win  with p = 2/3.
In other words, you have flipped the probability of
winning and losing.
7.2 Probability Theory
Red Flag of Caution!!
The amount of hideously horrendous applications of probability
and statistical reasoning is truly staggering – and can be life
threatening!
Why do all textbooks give dice throwing and
blackjack oriented exercises?
Because that is the only place where probabilities can be correctly
assessed! Beyond that – all bets are off (pun intended).
What does it mean to say that something is 90%
effective?
Does it mean that the outcome will be positive 95% of the time no
matter how many times it is tried? That is impossible! No
empirical data can become the basis for an unknown number of
trials.
Does it mean that 95 times out of 100 the outcome will be
positive? If so, then out of 1000 trials your chances of a negative
outcome are 40%; and out of 2000 your chances of a negative
outcome are 64%. Somehow that does not have the same ring as
95% effective!
Caveat::
To use probabilities in the real world you must:
1. Choose your set of events carefully and perspicaciously
2. Establish that all events have an equal probability, or
3. Abandon the equi-probability assumption and adjust the
event probabilities to reflect reality
4. Not make grand but bland probabilistic pronouncements
when further information germane to the case is readily
available.
Example:
You are on Rt. 222 and get behind a slow driver. You
both drive past 10 4-way intersections. At each the car in
front of you has 3 choices. Therefore, by the equi-probability
assumption, the chances that the car is still in front of you is
1/310 = 1/59049.
We know from experience that is NOT the correct
probability. Normal traffic patterns do not operate within the
equi-probability assumption.
All other things being equal;
In the absence of additional information
– the forgotten pre-nouncements.
Joel wants to become an actuary. There is a series of 5 tests
to take. For each test, only 85% of the persons pass it. What is his
probability of passing the 1st test? If he passes, what is his
probability of passing the 2nd test? If he passes the nth test what is
his probability of passing the (n+1)st test? Or is it?
First point – his probability of passing all 5 is (85/100)5 ≈
.4437, or < ½. But can we speak of his probability? Suppose 70
is a passing grade and he got 75 on the 1st exam? Is his probability
of passing the 2nd exam still 85%? What if we found a 2nd statistic
that tells us that only 10% of the persons who scored < 80 on an
exam went on to pass the next one? Is his probability now 10%?
What if we did not know his score on the 1st exam; would his
probability be 85% or 10%? What if he did not study at all and on
top of that was out partying the night before the exam; but is now
determined to buckle down and study? What is his probability
now? 10%? 85%? 90%?
Death by field trip – OR, a field day for philosophers
Stanford Encyclopedia of Philosophy discusses five major
interpretations of probability – classical, logical, frequency,
propensity and subjective. Wikipedia adds to the confusion with
its own meandering article. The kinds of problems that exist is
well illustrated in the example from van Fraasen, found in the
Stanford EoP, which starts out: “A factory produces cubes with
side-length between 0 and 1 foot; what is the probability that a
randomly chosen cube has side-length between 0 and 1/2 a foot?”
What we take away from this is that while some of the
mathematics is straightforward, what this means in the real world
and how it affects decision making can be a tough nut to crack (or,
as Latvians would say it: a cietais rieksts).