Applying set theory to probability
Dr. Ahmed Elmoasry
Applying set theory to probability
The mathematics we study is a branch of measure
theory
Probability is a number that describes a set
Flip a coin. Did it land a head or tails?
Give a lecture. How many students are seated in
the second row?
Walk to a bus stop. How long do you wait for the
arrival of a bus?
Applying set theory to probability
Waiting time at a bus stop, you may
consider:
The time of the day
The speed of each car
The wait, the horsepower, and gear of each
kind of bus used by the bus company.
The work schedule of each bus driver.
(some drivers drive faster)
Example 1
An experiment consist of the following
procedure, observation and model:
Procedure: Flip a coin and let it lands on a
table.
Observation: observe which side (head or
tail) faces you after the coin lands.
Model: Heads and tails are equally likely.
Note
Experiment consist of both procedure and
observations. Two experiments with the
same procedure but with different
observations are different experiments:
Ex. 1.2: Flip a coin three times. Observe
the sequence of heads and tails.
Ex. 2.3: Flip a coin three times. Observe
the number of heads.
Outcome, sample space , event
An Outcome of an experiment is any possible
observation of the experiment.
The Sample Space of an experiment is the finestgrain, mutually exclusive, collectively exhaustive
set of all possible outcomes.
An Event is a set of outcomes of an experiment.
Ex. 1.1 S={h, t}
Ex. 1.2: S={hhh, hht, hth, thh, htt, tht,tth,ttt}
Ex. 1.3: S={0,1,2,3}
Comparison between Set & probability
Set Algebra
Set
Universal Set
Element
Probability
Event
Sample Space
Outcome
Example
Suppose we roll a six sided die and observe
the number of dots on the side facing
upwards. We can label these outcomes
i=1,2,…, 6
Sample Space is
Example Subsets of S are
E1={Roll 4 or higher}={
}
E2={Roll is even}={
}
E3= {the roll is square of an integer}= {
}
Probability Axioms
A probability measure P[ ] is a function
that maps events in the sample space to real
number such that:Axiom 1: for any events A , P[A] 0.
Axiom 2: P[S] =1.
Axiom3 :  A1,A2,A3,…. & Ai∩Aj= 
P[A1A2  …]=p[A1]+p[A2]+p[A3]+….
Theorem
Theorem
Theorem
Theorem
Equally Likely outcomes
Theorem S={s1,s2,s3,…,sn}, each
outcomes is equally likely then
p[si] = 1/n 1 i n.
Example: for a die S= {1,2,3,4,5,6}
E1={4,5,6}
 p[E1]=
E2={2,4,6}
 p[E1]=
E3={1,4}
 p[E1]=
Theorem
The probability function (measure ) p[]
satisfies:p[]=
p[Ac]=
 A,B & A and B are disjoint , i.e. A∩B= 
p[AB]=p[A]+p[B]-p[A∩B]
if AB ,

p[A]  p[B]
Theorem
For any event A and event space
{B1,B2,…,Bm}P[A]= p[A∩Bi]
(AUB)c=Ac∩Bc
≠
Example
A company has model of telephone usage.
It classifies all calls as either long (L), if
they last more than 2 minutes, or brief (B).
It also observes whether calls carry voice
(V) , data (D), or fax(F)
The sample space S={}
V
F
D
L
0.3
0.15
0.12
B
0.2
0.15
0.08