Counting and Discrete Probability

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Counting and Discrete Probability

• Basic principles
• Permutations and Combinations
• Pigeonhole principle
• Discrete probability

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Two Basic Principles

• Multiplication Principle
• Addition Principle

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Multiplication Principle

• Suppose an activity can be performed in t
  successive steps

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Example

• strings of length 4 over Xa,b,c,d
• strings of length 4 over X that begins with a
• strings of length 4 over X that do not begin
  with a
• strings of length 4 over X with no repetition

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Example cardinality of a power set
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Example (Addition Principle)
How many six-bit strings begin with 101 or
001? six-bit strings begin with 101
2.2.28. six-bit strings begin with 001 8,
similarly. six-bit strings begin with 101 or
001 88 16.
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Example
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Inclusion-exclusion principle
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Example
7-bit strings which either begin with 0 or end
with 11 A 7-bit strings which begin with 0 B
7-bit strings which end with 11
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A permutation of n objects
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An r-permutation of n objects
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An r-combination of n objects is a set of r
elements selected from the n objects.

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Example
A coin is flipped 6 times. A possible outcome
H H T H T T
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Example
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Example
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Example
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Example
Given an n by n grid, how many ways are there to
go from (0,0) to (n,n), if in each step we can
either go right or go up? Eg. N3. ltUURURRgt is a
route. In general a route is represented by a a
sequence of length 2n with n Us and n Rs. To
form such a sequence choose n positions for U,
then fill in the remaining positions with R. So,
there are C(2n,n) routes.
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Example
routes from (0,0) to (n,n) as above, but we are
not allowed to go above the diagonal.
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Generalized Permutations and Combinations
How many different strings can be formed by
reordering letters in the word good? 4!/2.
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Theorem
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Proof
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Alternative proof
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Example
Eight different books to be given to Adams, Bill
and Calvin, where Adams gets four books and Bill
and Calvin each get two. First solution Choose
3 books for Adams C(8,4) Choose 2 books for
Bill C(4,2) Choose 2 books for Calvin
C(2,2) Hence, C(8,4)C(4,2)C(2,2) by
multiplication principle.
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Combinations with repetition
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Example
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Example
ways to distribute 15 identical books among 6
students.
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Binomial Coefficients
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Corollary
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Example
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Combinatorial identities from combinatorial
arguments
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Pigeonhole Principle
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Example
Ten persons have first names Alice, Bernard, and
Charles and last names Lee, McDuff and Ng. Then
at least two persons have the same first and last
names. Why? There are 9 possible names for the 10
persons 9 pigeonholes for 10 pigeons.
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Example
There are 20 persons in a room. Then at least
two persons have the same number of friends among
the 20 persons in the room.
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Example
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Example
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Example
If the decimal expansion of the quotient of two
integers is infinite, eventually some block of
digits repeat.
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Probability Theory
Developed in 17th century to analyze
games. Example Rolling a 6 sided fair die. The
probability that 3 appears. 1/6 The
probability that an even number appears.
3/61/2 Experiment Outcome Sample space Event
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Example Rolling a fair 6 sided die
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Example Rolling two fair dice
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Example
1000 microprocessors 20 defective Random
selection of 5 from the 2000
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General discrete probability
Outcomes need not be equally likely. Example
Rolling a loaded die 2-6 equally likely, but
1 three times as likely as any other
number.
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Definition of probability function
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Example Loaded die continued
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Theorem
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Example Two fair dice