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5.3
Continued
Conditional Probability
• Similar to a conditional distribution for
Chapter 4, that was the distribution of a
variable given that a condition is satisfied.
AGE
18-24
Married
3,046
Never Married 9,289
Widowed
19
Divorced
260
Total
12,614
25-64
65+
total
48,116
7,767
58,929
9,252
768
19,309
2,425
8,636
11,080
8,916
1,091
10,267
68,709
18,262
99,585
P(married) =
=
Would selecting a certain age
range change the probability?
1. P(married 18-24) =
* This is a Conditional Probability
The probability of one event (the woman
chosen is married) under the condition of
another event (she is between ages 18-24)
Read as “given the information that”
2. P(age 18-24 and married) =
3. P(age 18-24) =
• There is a relationship among these three
probabilities.
• The joint probability that a woman is both
age 18-24 and married is the product of
the probabilities that she is age 18-24 and
that she is married given that she is age
18-24.
• P(age 18-24 and married) =
P(age 18-24) x P(married age 18-24)
=
From the chart =
<as before>
General Multiplication Rule
• The Joint Probability that both of two
events A and B happen together can be
found by
P(A and B) = P(A) P(B A)
 P(B A) is the conditional probability that B
occurs given the info that A occurs
• With algebra we get Definition of
Conditional Probability
 When P(A) > 0 , the conditional
probability of B given A is
P(B A) = P(A and B)
P(A)
• Ex. 1) What is the conditional probability
that a woman is a widow, given that she is
at least 65 years old?
P(at least 65) =
P(widowed and at least 65) =
=
=
The conditional probability
P(widowed at least 65) = P(widowed and at least 65)
P(at least 65)
=
 Check results from actual table
=
=
• The Intersection of any collection of events
is the event that all the events occur
• The Multiplication Rule extends to the
probability that all of several events occur.
P(A and B and C) = P(A) x P(B
A) x P(C A and B)
• Ex.) 5% of male high school basketball/
baseball/ football players go on to play in
college
• Of these, 1.7 % enter majors. About 40%
of the athletes who compete in college and
reach the pros have a career of 3+ years.
A = {compete in college}
B = {compete in professional}
C = {pro. Career 3+ years}
• What is the probability that a high school
athlete competes in college and then goes
on to have a professional career in 3+
years?
P(A) =
P(B A) =
P(C|A and B) =
P(A and B and C) = .
(3 out of 10,000!)
Tree Diagram
Professional
College
A
B
B
High School
Athlete
c
B
A
c
B
c
• There are 2 disjoint paths to B (play
professionally)
• By the addition rule, P(B) = sum of those
probabilities
• Probability of reaching B through college is
P(B and A) = P(A) P(B A) =
• Probability of reaching B without college is
c
c
c
P(B and A ) = P(A ) P(B A ) = .
• Final result P(B) = .00085 + .000095 = .000945
= 9/10,000 High school athletes play
professionally
Tree diagrams combine the
addition and multiplication rules.
• Mult. Rule states: Probability reaching the
end of any branch is the product of the
probabilities written on segments.
• The product of any outcome, such as the
event B, an athlete reaches professional
sports is then found by adding probabilities
of all branches that are a part of that
event.

Two events A & B, both pos. probability
and independent if,
P(B A) = P(B)