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BA 1605
6.2 Joint, Marginal, and Conditional Probability/ 6.3 Probability Rules and Trees
Compound events are formed by the composition of 2 or more events. In the next 2
sections we will look at combining 2 or more events in certain ways.
Union and Intersection of Two Events
 A or B (denoted A U B) is the event that A occurs or B occurs or they both occur.
This is known as the “union” of two events.
 A and B (denoted A ∩ B) is the event that both A and B occur (ie. Common to
both A and B). This is known as the intersection of A and B.
 Probability of the intersection of two events are also known as joint probabilities.
 The Addition Rule says P A  B  P A  PB  P A  B .
 We will examine the Multiplication Rule later, which helps us find the P(A∩B)
Example
Throw a die and observe the number of dots. Let A be the event that an odd number is
observed, B the event that a number less than 5 is observed, C the event that 6 is observed
and D the event that 3 is observed.
A U B = {1,2,3,4,5}
A U C = {1,3,5,6}
B U C = {1,2,3,4,6}
A U D = {1,3,5}
B U D = {1,2,3,4}
D U C = {3,6}
Repeat for the intersections
We say 2 events are mutually exclusive or disjoint if they cannot occur at the same time.
If this is the case, their intersection is empty, or called the null set.
When A and B are mutually exclusive A ∩ B is empty and P A  B  0 and
P A  B  P A  PB
We can use Venn Diagrams to illustrate the relationships between events.
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Example
In a survey of 1000 students it was found that
300 were taking Statistics
350 were taking Math
200 were taking Accounting
100 were taking Statistics and Accounting
150 were taking Math and Accounting
200 were taking Statistics and Math
75 were taking Statistics, Math and Accounting
Construct a Venn Diagram and find the probability that a student is taking
a) Statistics, Math and Accounting
b) Statistics, Math or Accounting
c) Statistics, but not Math or Accounting
d) Not taking Statistics, Math or Accounting
e) Taking Accounting and Statistics but not Math
f) Statistics
g) Not taking Math
h) Taking Statistics or Math
Example
On New Year’s Eve, the probability of a person driving while intoxicated is 0.32, the
probability of having a driving accident is 0.09 and the probability of a person having a
driving accident while intoxicated is 0.06. What is the probability of a person driving
while intoxicated or having a driving accident?
(0.35)
We can use a contingency or two way table to help organize data to find probability:
Example
In a hospital unit there are 8 nurses and 5 physicians. Seven nurses and three physicians
are females. If a staff person is selected, find the probability that the subject is a nurse or
male.
(10/13)
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Conditional Probability
Two events are often related such that the probability of one event occurring is dependent
on whether or not the other event has occurred.
The conditional event, A given B (denoted A | B) , means the event that A occurs given B
has already occurred.
Example:
A – rain occurs, B – cloudy day, P A | B   P( A | B)
We call A and B dependent events.
The conditional probability of A given B is found by
Example:
P A | B  
P A  B 
PB 
Draw a single card out of a standard deck. Let A be the event that a red
card is drawn, B a spade is drawn, C the event that an ace is drawn: Find
the probability of:
a) A | B
b) A | B
c) A | C
d) C | A
e) B | B
f) B | B
Sometimes we do not need a formula
Example:
Draw 2 cards from a standard deck without replacement. Let A be red
card, B spade. Find the probability of:
a) B2 | A1
b) B2 | B1
c) A2 | A1
d) A2 | B1
Example:
Draw a number out of a hat containing slips of paper numbered 1 to 10.
Let A be the event that 7 or 8 is drawn and B the event that an odd number
is drawn. Find the P(A), P(B), P(A|B).
Two events, A and B, are said to be independent when the occurrence of one event has no
effect on the occurrence of the other event. Events A and B are independent iff
P(A|B)=P(A) and P(B|A)=P(B)
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Example:
In a survey of 10 consumers,
4 indicated they used product A
5 indicated they used product B
2 indicated they used both products
Find the probability that a randomly selected customer
a) uses A or B
b) uses A and B
c) uses B
d) uses A given he uses B
e) does not use B
f) Are A and B independent events?
Example:
The table below gives the results of a survey on the drinking and smoking
habits of university students.
Drinks Beer
Smokes
Doesn’t Smoke
Totals
315
585
900
Doesn’t Drink
Beer
165
135
300
Find the probability that
a) someone in this group smokes
(0.40)
b) a student smokes, if he or she is a beer drinker (0.35)
c) someone drinks
(0.75)
d) a randomly selected student drinks, given he or she smokes
Totals
480
720
1200
(0.65)
Note: The events the student drinks beer and the student smokes are not independent
events.
Example: If A and B are mutually exclusive events with P(A) > 0 and P(B) > 0. Are A
and B independent events?
Answer: no
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Multiplication Rule:
P A  B  P A  PB | A
or
P A  B  P A | B  PB
Note: If A and B are independent events then P A  B  P A  PB
Example:
Find the probability of getting all tails in
a) 2 tosses of a fair coin.
b) 3 tosses of a fair coin
c) n tosses of a fair coin
Example:
Find the probability of rolling three 6’s when tossing a fair die three times.
Example:
Drawing 2 cards from a standard deck, find the probability of getting both
aces if the first card is a) not replaced (1/221)
b) replaced (1/169)
We can use tree diagrams in helping us answer questions about probability.
Example:
If a couple decides to have 3 children find the probability that
a) all are girls
(1/8)
b) at least one child is a girl
(7/8)
c) exactly one of the three is a girl (3/8)
Example:
We receive a shipment of 10 thermometers. Unknown to us, 3% are
defective. We plan to randomly select 2 thermometers and test for defects.
Find the probability of obtaining
a) a defective second thermometer, given the first was good
(1/3)
b) both good
(7/15)
c) a good item first and a defective item second (7/30)
d) both defective (1/15)
e) a good item and a defective item
(7/15)
Pages 166 – 170, 175 - 177