Sometimes to calculate the probability of
an event directly is tedious, while it is
easier to obtain the probability of the
complement of this event.
Example 14. Probability of Employee Selection
A Congresswoman wants to hire four assistants
from eight candidates. Five candidates are men,
and three are women. If every combination of
candidates is equally likely to be chosen, what is
the probability that at least one women will be
Example 15. Born the Same Day?
(A great question for a party!)
There are 29 students in class Econ390. What
is the probability that at least two students
have the same birthday?
Consider a pair of events A and B. Suppose we are
concerned about the probability of A given that B has
The idea is that the probability of an event is often
affected by knowledge of circumstances.
For example, if we randomly select someone from the
general population, the probability of getting a male is 0.5,
but if we know that the selected person frequently
changes TV channels with a remote control, what is the
probability in this case?
Conditional Probability of A given B is denoted
as P(A l B)
and defined as
P(A l B) = P (A ∩ B) / P(B) (Assumes P(B)≠0)
P(B l A) = P (A ∩ B) / P(A) (Assumes P(A)≠0)
Use relative frequencies to help us understand conditional probability.
From the definition of condition probability, we can obtain the
Multiplication Rule of Probability
P (A ∩ B) = P(B l A) P(A)
P (A ∩ B) = P(A l B) P(B)
Example 1. Product Choice
A hamburger chain found that 75% of all customers
use mustard, 80% use ketchup, and 65% use both.
If you are given the information that a randomly
selected customer uses ketchup, what is the probability
that she/he also uses mustard?
Event A and event B are said to be statistically independent
if and only if
P(B | A) = P(B) or P(A | B ) =P(A)
which means the probability of B occurring doesn’t change even
after given the information that A has occurred.
If event A and event B are statistically independent, the
multiplication Rule is
P (A ∩ B) = P(B | A) P(A) = P(B)P(A)
There are three ways to test if event A and B are statistically
Example 2. Probability of College Degrees
Suppose that 48% of all bachelor degrees in a
particular country are obtained by women and that 17% of
all bachelor degrees are in business. Also, 6% of all
bachelor degrees go to women majoring in business. Are
the events “Bachelor degree holder is a women” and
“Bachelor degree is in business” statistically independent?
Statistically Independent Events ≠ Disjoint Events
A and B independent means the information of A
occurs or not does not affect the probability of B
If A and B are disjoint, the information of A occurring
indicates that B does not occur.
So if A and B are disjoint, they must be statistically
If A and B are disjoint, P (A ∩ B) =0
If A and B are independent, P (A ∩ B) =P(A)P(B)
Applying Multiplication Rule
Example 3. Foul-ups
A mail-order firm considers three possible foul-ups in filling an
A: The wrong item is sent.
B: The item is lost in transit.
C: The item is damaged in transit.
Assume that event A is independent of both B and C, and that
events B and C are mutually exclusive. The individual event
probabilities are P(A)=0.02, P(B)=0.01, P(C)=0.04. Find the
probability that at least one of these foul-ups occurs for a
randomly chosen order.
Hint: P(A U B U C)=
P(A) + P(B) + P(C) - P(A ∩ B) – P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)
Example 4. On-Job Training
A corporation provides separate classes in reading
and practical mathematics for its employees. 40% of
workers signed up for the reading classes, and 50%
for the practical mathematics classes. Of those signing up
for the reading classes, 30% signed up for the
(1) What is the probability that a randomly selected worker
signed up for both classes?
(2) What is the probability that a randomly selected worker
who signed up for the mathematics classes also signed
up for the reading classes?
(3) What is the probability that a randomly chosen worker
signed up for at least one of these two classes?
(4) Are the events “Signs up for reading classes” and
“Signs up for mathematics classes” statistically
Example 5. Loan Default
A bank classifies borrowers as high-risk or low-risk. Only 15%
of its loans are made to those in the high-risk category. Of all its
loans, 5% are in default, and 40% of those in default are to high-
risk borrowers. What is the probability that a high-risk borrower will
A=a loan is made to high-risk category
B=a loan is in default
P(A)=0.15, P(B)=0.05, P( A | B )=0.4, P(B | A)=?