Ch2.3-2.4

2.3 Counting TechniquesProduct 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 n1n2.

** Note that this generalizes to k elements (k – tuples)

Permutations

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

, ( 1) ... ( 1) k nP n n n k

Ch2.3-2.4

Combinations

Given a set of n distinct objects, any unordered subset of size k of the objects is called a combination.

Notation: , or k n

nC

k

!

! !

n n

k k n k

Ch2.3-2.4

2.4 Conditional ProbabilityFor any two events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is defined by

Which can be written:

|P A B

P A BP B

|P A B P B P A B

Ch2.3-2.4

2.4

The Law of Total Probability

Let the events A1, A2,…, Ak be mutually exclusive and exhaustive events. The for any other event B,

1

( | ) ( )k

i ii

P B P B A P A

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2.4

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

1

||

|

j jj k

i ii

P A P B AP A B

P A P B A

1,2...,j k

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2.4

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.

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2.4

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?