4.2 Probability Models
We call a phenomenon random if
individual outcomes are uncertain but
there is nonetheless a regular distribution
of outcomes in a large number of
repetitions.
Experiments
Consider tossing a fair coin 5 times. Let H
represent obtaining a head and T represent
obtaining a tail. Then one possible outcome is
HHTHT.
This is an example of an experiment; that is, any
activity that yields a result or an outcome.
A survey with yes/no/undecided outcomes is an
experiment.
Sample Space
If we consider the coin tossing experiment mentioned
above, the possible outcomes are HHHHH, HHHHT,
HHHTH, HHTHH, HTHHH, THHHH, HHHTT, HHTHT,
HTHHT, etc.
The collection of all possible distinct outcomes that can
occur when an experiment is performed is called the
sample space.
The sample space must have the property that when the
experiment is performed, exactly one of these outcomes
must occur.
We may wish to give the sample space a name (say S)
and we usually write the sample space inside of
brackets. In the case of a single toss of a coin, the
sample space would be written S={H,T}.
Events
An event is a subset of the sample space.
An event can be a single outcome or
several. In the case where the event is a
single outcome, we call this a simple
event.
If we let S={H,T}, then an event can be H
or TTTTH. If we let S={HHHHH,
HHHHT,HHHTH,HHTHH, …} then the
event TTTTH is simple.
What is probability?
Suppose that A is some
event. The probability of
A, denoted P(A), is the
expected proportion of
Number of outcomes in A
occurrences of A if the
P( A) 
Number of outcomes in S
experiment were to be
repeated many times. In
other words, if S is the
sample space and A is an
event then
Suppose we roll a fair die 2 times. What is the
probability that the product of the two rolls is
divisible by 3?
This requires finding not only the size of the
sample space, but also the number of outcomes
that satisfy our condition.
We’ll use brute force.
S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,
3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6
),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3)
,(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
Listing out all possibilities and then
counting one by one is admittedly not a
very good way of doing things. Much of
what we discuss in the sections on
probability will therefore be concerned with
convenient was to count.
How many distinct strings comprised of
three letters can be made out of the 26
letters of the alphabet?
Multiplication Rule
This leads us to the Fundamental Principle
of Counting or the Multiplication Rule: If
Task 1 can be performed in n ways and
Task 2 can be performed in m ways, then
Task 1 and Task 2 can be performed
together in nm ways.
How many ways are there to toss a fair die
5 times with each roll showing a different
number than the previous rolls?
We have looked at P(A); that is, the probability
of a single event A taking place e.g. when rolling
a die 5 times, what is the probability that the sum
of the 5 rolls will be divisible by 4? Call this
event A.
Now let B be the event of rolling a die 5 times
and obtaining an even number on every roll.
What is the probability of both A and B
happening? Or what is the probability that at
least one of A or B happens? Or what is the
probability that A does NOT happen? These are
compound events.
Language, Truth, and Logic
Let A and B be events from a sample space S.
A or B is the event that either A occurs or B
occurs or both. This event is usually denoted
AυB, read A union B or A or B.
A and B is the event that both A occurs and B
occurs at the same time. This event is usually
denoted A∩B, read A intersect B or A and B.
The complement of A is the event that an
outcome in S that is NOT in A will occur. The
event is denoted Ā, read the complement of A.
We may illustrate these with Venn diagrams.
Disjoint Events
Consider a deck of 52 playing cards (13 spades,
13 clubs, 13 hearts, 13 diamonds) and consider
an experiment where we draw one card at
random. Let E be the event of drawing a black
card with an even value (2,4,6,8,10) and D be
the event of drawing a diamond. What is the
probability of E υ D (E or D)?
Events A and B are disjoint events if they can
not occur together when the experiment is
performed.
The Addition Rule
Suppose A and B are disjoint events.
Then P(A or B)=P(A)+P(B). If C is a third
event which is mutually exclusive with both
A and B, then P(A or B or C)=
P(A)+P(B)+P(C) etc.
If a pair of dice is rolled, find the probability
of rolling a double or getting a sum of 9.
Independent events
Consider an experiment where a card is
randomly drawn from a deck (52 cards),
recorded, replaced, the deck is shuffled,
and another card is drawn. Let A be the
event of drawing a face card (king, queen,
jack) on the first draw and B be the event
of drawing a red card on the second draw.
What is the probability of A and B?
Events A and B are said to be independent if the
occurrence of one has no effect on the
probability of the occurrence of the other.
Suppose A and B are independent events. Then
P(A and B)=P(A)P(B).
Suppose two fair dice are rolled. What is the
probability that the first one will show an even
number and the second one will show an odd
number?
The Monty Hall Problem
Complementary Events
Let A be an event. Then Ā is any event in
S which is not in A. Hence P(A)+P(Ā)=1
and so P(Ā)=1-P(A). This fact can often
make problems easier.
Suppose we roll three 20-sided dice (we’re
playing D & D). What is the probability
that no single die will show a 20?