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Transcript Ch2.3-2.4

2.3 Counting Techniques
Product Rule
If the first element or object of an ordered pair can be used
in n1 ways, and for each of these n1 ways the second
can be selected n2 ways, then the number of pairs is
** Note that this generalizes to k elements (k – tuples)
Any ordered sequence of k objects taken from a set of n
distinct objects is called a permutation of size k of the
objects. Notation: Pk,n
Pk ,n  n(n  1)  ...  ( n  k  1)
Given a set of n distinct objects, any unordered subset of
size k of the objects is called a combination.
Notation:  n  or C
 
 
k 
k ,n
k 
k ! n  k  !
2.4 Conditional Probability
For any two events A and B with P(B) > 0, the conditional
probability of A given that B has occurred is defined by
P  A | B 
P  A  B
P  B
Which can be written:
P  A  B  P  B  P  A | B
The Law of Total Probability
Let the events A1, A2,…, Ak be mutually exclusive and
exhaustive events. The for any other event B,
P  B    P( B | Ai ) P( Ai )
i 1
Bayes’ Theorem
Let A1, A2, …, An be a collection of k mutually exclusive and
exhaustive events with P(Ai) > 0 for i = 1, 2,…,k. Then
for any other event B for which P(B) > 0 given by
P Aj | B 
  
P Aj P B | Aj
 P  Ai  P  B | Ai 
i 1
j  1, 2..., k
Example 3
A blood test detects a certain disease 99% of the time
when the disease is present. When a healthy person is
tested, however, there is a 2% that the test will say he or
she has the disease. Suppose 0.5% of the population
has the disease. Find the conditional probability that a
randomly tested person has the disease given his or her
test says that he or she has it.
Example 4
Three different machines M1, M2, M3 are used to make a
large batch of similar items. Suppose 20% of the items
are produced by M1, 30% by M2, 50% by M3. Suppose
also that 1% of the items produced by M1 are defective,
as are 2% of those produced by M2 and 3% of those
produced by M3. If one item is selected at random from
the entire batch and is found to be defective, what is the
probability that it was produced by M2?