ST361: Ch5.3 Conditional Probability and Independence
-----------------------------------------------------------------------------------------------------------Topics:
§5.3: Conditional probability
Multiplicative law
Independent events
Calculating probabilities
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 Conditional Probability:
 Consider the distribution of blood type in a certain population:
Blood
Type
O
A
B
AB
Total
Total # of
people
70
50
50
30
200
Let “O” denote the event of randomly selecting a person whose blood type is O.
P(O) = ? How about P(A) and P(AB)?
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 Assume that the information of sex distribution is also available:
Blood
Type
O
A
B
AB
Total
Total # of
people
70
50
50
30
200
 More info
Blood
Type
O
A
B
AB
Total
Male
Female
37
27
26
16
106
33
23
24
14
94
Total # of
people
70
50
50
30
200
Then we can calculate the following probabilities. Denote “Male” the event that
randomly select a person who is male, and “Female” the event that randomly
select a person who is female.
 Probability of randomly select a person who is male
 Probability of randomly select a person who is male and blood type “O”
 Even the probability that selecting a blood type “O” person among males
 Can classified these probabilities into
(1) ___________________________, i.e., the probability of one event, ignoring
any information about the other event.
Ex. Row margins:
Column margins:
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(2) ___________________________, i.e., the probability of two events considered jointly,
Ex. P( Male and O ) =
P( AB and Female ) =
(3) __________________________, i.e., the probability of one event given
(conditioning) on another one event has occurred.
Ex. Among males, the probability of being type “O” =
Such probability is denoted by
Ex. P( AB | Female) =
P(Female | O ) =
 Conditional distribution is all about reducing the space of interest from the
___________________________ to _____________________________
 The conditional probability of A given that event B has occurred is
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Ex. A student is randomly selected from a class where 35% of the class is lefthanded and 50% are sophomores. We further know that 5% of the class
consists of left-handed sophomores. Given that a randomly selected student
is a sophomore, what is the probability that he/she is left-handed?
What we know:
What we want:
Solve:
Ex. A certain system can experience 2 difference types of defects. Let Ai , i=1,2
denote the event that the system has a defect of type i. Suppose that
P A1   0.15, P A2   0.10 , and P A1 and A 2   0.08 . If the system has a type 1
defect, what is the probability that it has a type 2 defect?
What we know:
What we want:
Solve:
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 Multiplicative Law:
Ex. Six balls in a basket: 2 white, 3 blue and 1 yellow. Two balls are drawn
randomly one after the other. What is the probability that the first ball is
white, and the second ball is white?
What we know:
What we want:
Solve:
Method 1: to solve by applying the multiplicative law
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Method 2: to solve this problem using a Tree Diagram.
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Ex. A store sells 2 different brands of DVD players. Of its DVD player sales, 60% are
brand A (less expensive) and 40% are brand B. Each manufacturer offers a 1-yr
warranty on parts and labor. It is known that 25% of brand A’s DVD players require
warranty repair work, whereas 10% for brand B.
(a) What is the probability that a randomly selected purchaser has bought a brand A
DVD player that will need repair while under warranty?
What we know:
What we want:
Solve:
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(b) Assume the probability that a randomly selected purchaser has a DVD player
that will need repair while under warranty is 0.19. Now if a customer returns to
the store with a DVD player that need warranty repair work, what is the
probability that it is brand A? Brand B?
What we know:
What we want:
Solve:
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 Independent events:
(1) Events A and B are independent events if ___________________________
_____________________________________________________________
That is,
Therefore, when events A and B independent,
P( A and B ) =
(2) For k independent events A1 , A2 ,..., Ak ,
P A1 and A2 and .... and Ak  =
Ex. A system consists of 3 components, as shown in the picture. The components
work or fail independently of one another, and each component works with
probability 0.9. What is the probability that the entire system will work?
(a)
1
2
3
What we know:
What we want:
Solve:
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(b)
2
1
3
What we know:
What we want:
Solve:
Ex. Two independent services A and B are used to deliver a document. The
probability of on-time delivery for service A is 0.8, and for service B is 0.7.
What is the probability that the document being delivered on time?
What we know:
What we want:
Solve:
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