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381
Probability-II
(The Rules of Probability & Counting Rules)
QSCI 381 – Lecture 8
(Larson and Farber, Sects 3.3+3.4)
Independent Events-I
381
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Two events are said to be
if
the occurrence of one does not affect the
probability of the other, i.e. the probability of
the event B is the same as the probability of
the event B given A.
Two events are independent therefore if:
P( A)  P( A | B);
P( B)  P( B | A)
Note: Knowing that B has occurred doesn’t impact the
probability of whether A will occur if A and B are independent.
Independent Events-II
381
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Which of these events are independent:
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Being male and playing football.
In this class and enjoying tennis.
Being in this class and knowing about
statistics.
The Multiplication Rule-I
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The probability that two events A and B will
occur in sequence is:
P( A and B)  P( A  B)  P( A).P( B | A)

If A and B are independent, then the
multiplication rule becomes:
P( A and B)  P( A).P( B)
The Multiplication Rule-II
(Example)
381
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Consider evaluating the probability of extinction of a
species that consists of 10 sub-populations when the
probability of an individual sub-population becoming
extinct is 0.1.
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What is the probability of the species becoming extinct when
the probability of extinction of one sub-population is
independent of that of any of the others?
What is the probability of extinction when the process
leading to extinction is common to all sub-populations?
Hint: Write down what you know (look for any events
that are conditional on others).
Mutually Exclusive Events
381
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Two events A and B are
if A and B cannot occur at the same time
Questions to assess whether two events A
and B are mutually exclusive:
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Can A and B occur at the same time?
Do A and B have outcomes in common?
Can you think of some mutually exclusive
events?
The Additive Rule
381
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The probability that events A or B will occur
is:
P( A or B)  P( A  B)  P( A)  P( B)  P( A and B)

If events A and B are mutually exclusive,
then:
P( A or B)  P( A)  P( B)
381
Using the Rules of Probability
(Example)
Blood Type
Rh
factor
O
A
B
AB
Total
Positive
156
139
37
12
344
Negative
28
25
8
4
65
Total
184
164
45
16
409
The above table is based on data for 409 randomly selected blood donors
• What is the probability that a donor has type O or type A blood?
• What is the probability that a donor has type B blood and is Rh-negative?
• What is the probability that a donor has type A blood and is Rh-negative?
• What is the probability that a donor is Rh-positive given he / she has
blood type O?
381
Review of Concepts and Formulae
P( A) :probability of event A
P( A ')  1  P ( A) :probability of the compement of the event A
P( A)  P( A and B)  P( A and B ') :decomposition of the probability of event A
P( B | A)  P( A and B) / P( A) :conditional events
P( B | A)  P( B) :independent events
P( A and B)  P( B | A).P( A) :multiplication rule
P( A and B)  P( A).P( B) :independent events
P( A or B)  P( A)  P( B)  P( A  B ) :addtion rule
P( A or B)  P( A)  P( B) :mutually exclusive events
The Fundamental Counting Principle-I
381
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If one event can occur in m ways and a
second event can occur in n ways, the
number of ways in which the two
events can occur in sequence is m x n.
The Fundamental Counting Principle-II
(Examples)
381
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You are sampling salmon: there are four watersheds,
three streams in each watershed and four species in
each stream. How many ways to select one
watershed, species, and stream?
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4 x 3 x 4 = 48. Check this by listing them.
You are ageing fish. The sample is 10 animals and
the fish are numbered 1,2 ..10. How many ways are
there to age 4 of the 10 fish (fish are aged once and
once only).
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How many ways to select the first fish, the second fish….