Sixth lecture
Concepts of
Probabilities
Random Experiment
• Can be repeated (theoretically) an infinite
number of times
• Has a well-defined set of possible outcomes.
• Result cannot be predetermined.
Example 1:
• Selecting 20 people at random and counting
how many people are left-handed.
• Tossing a coin until the first tail appears.
How can we describe random experiments using a
mathematical model?
We use three building blocks: a sample space, a set
of events and probability.
Definition 1 (Sample Space):
The set of all possible outcomes of a statistical
experiments is called the sample space and
represented by the symbol S.
Example 2:
Select 20 people at random and count # of lefthanders
S= {0, 1, 2, … ,20}
Example 3:
Consider the experiment of tossing a die. If we
are interested in the number that shows on the
top face, describe the sample space?
S= {1, 2, 3, 4, 5, 6}
If we are interested only in whether the number
is even or odd, the sample space is simply
S= {even, odd}
Definition 2 (An event is) :
The subset of a sample space. Donated by a
capital letter.
Example 4:
Tossing two dice and A is the event that the sum
of the two faces is ≥ 10.
S= {(i, j), 1≤ i, j ≤ 6}
A= {( 4, 6) , (5, 6), (6, 6), (6, 4), (6, 5), (6,6)}.
The complement:
The complement of an event A with respect to S
is the subset of all elements of S that are not in
A. We denote the complement of A by the
symbol Aˊ.
The intersection
The intersection of two events A and B,
denoted by the symbol A ∩ B , is the event
containing all elements that are common to A
and B.
A∩B
Definition 3:
Two events A and B are mutually exclusive or
disjoint, if A ∩ B= Ø, that is, if A and B have
no elements in common.
Definition 4: The union of the two event A and
B denoted by the symbol A U B is the event
containing all the elements that belong to A or
B or both.
• Example 5: (set operations)
let S= {0, 1, 2, …, 7}, A= {0, 2, 6}, B= {1, 2}
and C= {0, 6}.
Find A ∩ B, A U B, B ∩ C, Aˋ.
• A ∩ B = {2}
• A U B = {0, 1, 2, 6}
• B ∩ C= Ø
• Aˋ= {1, 3, 4, 5, 7}
Some important results related to the set operation:
• A ∩ Ø = Ø.
• A U Ø = A.
• A ∩ Aˊ = Ø.
• A U Aˊ = S.
• Sˊ = Ø.
• Ø ˊ= S.
• (Aˊ)ˊ = A.
• (A ∩ B)ˊ = Aˊ U Bˊ
• (A U B)ˊ = Aˊ ∩ Bˊ
Probability functions
Want to assign a probability to an experiment's
outcome (and in general to events).
• Let A be an event defined on sample space S
• P(A) denoted the probability of A occurring
• P is the probability function
Definition 5:
The probability of an event A is the sum of the
weights of all sample points in A. Such that
Axiom 1: 0 ≤ P(A) ≤ 1,
Axiom 2: P(S)= 1 and P(Ø) = 0,
Axiom 3: if A, B, C, …. Is a sequence of
mutually exclusive events then
P(A U B U C U …)= P(A) + P(B) + P(C)+ ….
Example 6:
A coin is tossed twice. What is the probability
that at least one head occurs?
Solution :
The sample space is
S= {HH, HT, TT, TH}
Let A be the event that at least a head occurs.
If we assume the coin is balanced all outcomes
would be equally likely to occur, therefore
The probability for any element to occur is ¼
And P(A)= ¾.
Theorem 1:
If an experiment can result in any one of N
different equally likely outcomes, and if
exactly n of these outcomes correspond to
event A, then the probability of event A is
P(A)= n/N
Example 7:
Consider the experiment of tossing a die. What is
the probability that we get an even number?
Solution:
The sample space is S= {1, 2, 3, 4, 5, 6}
Let A be the event that we get an even # A={2,4,6} ,
because the outcomes all equally likely to
happened, so we will be able to apply the
previous theorem.
N = 6, n= 3 therefore P(A) = 3/6=1/2.
Additive Rules
Theorem 2:
If A and B are two events, then
P(A U B)= P(A) +P(B) – P(A ∩ B).
Result 1:
• If A and B are mutually exclusive, then
P(A U B)= P(A) +P(B)
• If A₁, A₂, A₃, … Aᵣ is a partition of sample space S then
P(A₁ U A₂ U A₃U … U Aᵣ )= P(A₁) + P(A₂)+ P(A₃) + …
+P(Aᵣ)=P(s)=1.
Theorem 3:
If A and Aˊ are complementary events then
P(A)+ P(Aˊ)= 1.
Example 8:
What is the probability of getting a total of 7 or
11 when a pair of fair dice are tossed?