Discrete Mathematics Lecture 7

1
Discrete MathematicsLecture 7
Harper Langston New York University
2
Poker Problems

• What is a probability to contain one pair?
• What is a probability to contain two pairs?
• What is a probability to contain a triple?
• What is a probability to contain royal flush?
• What is a probability to contain straight flush?
• What is a probability to contain straight?
• What is a probability to contain flush?
• What is a probability to contain full house?

3
Combinations with Repetition

• An r-combination with repetition allowed is an
  unordered selection of elements where some
  elements can be repeated
• The number of r-combinations with repetition
  allowed from a set of n elements is C(r n 1,
  r)
• Soft drink example

4
Algebra of Combinations and Pascals Triangle

• The number of r-combinations from a set of n
  elements equals the number of (n
  r)-combinations from the same set.
• Pascals triangle C(n 1, r) C(n, r 1)
  C(n, r)
• C(n,r) C(n,n-r)

5
Binomial Formula

• (a b)n ?C(n, k)akbn-k
• Examples
• Show that ?C(n, k) 2n
• Show that ?(-1)kC(n, k) 0
• Express ?kC(n, k)3k in the closed form

6
Probability Axioms

• P(Ac) 1 P(A)
• P(A ? B) P(A) P(B) P(A ? B)
• What if A and B mutually disjoint?(Then P(A ? B)
  0)

7
Conditional Probability

• For events A and B in sample space S if P(A) ¹ 0,
  then the probability of B given A is P(A B)
  P(A ? B)/P(A)
• Example with Urn and Balls- An urn contains 5
  blue and

8
Conditional Probability Example

• An urn contains 5 blue and 7 gray balls. 2 are
  chosen at random.- What is the probability they
  are blue?- Probability first is not blue but
  second is?- Probability second ball is blue?-
  Probability at least one ball is blue?-
  Probability neither ball is blue?

9
Conditional Probability Extended

• Imagine one urn contains 3 blue and 4 gray balls
  and a second urn contains 5 blue and 3 gray balls
• Choose an urn randomly and then choose a ball.
• What is the probability that if the ball is blue
  that it came from the first urn?

10
Bayes Theorem

• Extended version of last example.
• If S, our sample space, is the union of n
  mutually disjoint events, B1, B2, , Bn and A is
  an even in S with P(A) ¹ 0 and k is an integer
  between 1 and n, thenP(Bk A)
  P(A Bk) P(Bk) .
• P(A B1)P(B1) P(A
  Bn)P(Bn)
• Application Medical Tests (false positives,
  etc.)

11
Independent Events

• If A and B are independent events, P(A ? B)
  P(A)P(B)
• If C is also independent of A and B P(A ? B ? C)
  P(A)P(B)P(C)
• Difference from Conditional Probability can be
  seen via Russian Roulette example.

12
Generic Functions

• A function f X ? Y is a relationship between
  elements of X to elements of Y, when each element
  from X is related to a unique element from Y
• X is called domain of f, range of f is a subset
  of Y so that for each element y of this subset
  there exists an element x from X such that y
  f(x)
• Sample functions
• f R ? R, f(x) x2
• f Z ? Z, f(x) x 1
• f Q ? Z, f(x) 2

13
Generic Functions

• Arrow diagrams for functions
• Non-functions
• Equality of functions
• f(x) x and g(x) sqrt(x2)
• Identity function
• Logarithmic function

14
One-to-One Functions

• Function f X ? Y is called one-to-one
  (injective) when for all elements x1 and x2 from
  X if f(x1) f(x2), then x1 x2
• Determine whether the following functions are
  one-to-one
• f R ? R, f(x) 4x 1
• g Z ? Z, g(n) n2
• Hash functions

15
Onto Functions

• Function f X ? Y is called onto (surjective)
  when given any element y from Y, there exists x
  in X so that f(x) y
• Determine whether the following functions are
  onto
• f R ? R, f(x) 4x 1
• f Z ? Z, g(n) 4n 1
• Bijection is one-to-one and onto
• Reversing strings function is bijective

16
Inverse Functions

• If f X ? Y is a bijective function, then it is
  possible to define an inverse function f-1 Y ? X
  so that f-1(y) x whenever f(x) y
• Find an inverse for the following functions
• String-reverse function
• f R ? R, f(x) 4x 1
• Inverse function of a bijective function is a
  bijective function itself

17
Pigeonhole Principle

• If n pigeons fly into m pigeonholes and n gt m,
  then at least one hole must contain two or more
  pigeons
• A function from one finite set to a smaller
  finite set cannot be one-to-one
• In a group of 13 people must there be at least
  two who have birthday in the same month?
• A drawer contains 10 black and 10 white socks.
  How many socks need to be picked to ensure that a
  pair is found?
• Let A 1, 2, 3, 4, 5, 6, 7, 8. If 5 integers
  are selected must at least one pair have sum of 9?

18
Pigeonhole Principle

• Generalized Pigeonhole Principle For any
  function f X ? Y acting on finite sets, if n(X)
  gt k N(Y), then there exists some y from Y so
  that there are at least k 1 distinct xs so
  that f(x) y
• If n pigeons fly into m pigeonholes, and, for
  some positive k, m gtkm, then at least one
  pigeonhole contains k1 or more pigeons
• In a group of 85 people at least 4 must have the
  same last initial.
• There are 42 students who are to share 12
  computers. Each student uses exactly 1 computer
  and no computer is used by more than 6 students.
  Show that at least 5 computers are used by 3 or
  more students.

19
Composition of Functions

• Let f X ? Y and g Y ? Z, let range of f be a
  subset of the domain of g. The we can define a
  composition of g o f X ? Z
• Let f,g Z ? Z, f(n) n 1, g(n) n2. Find f
  o g and g o f. Are they equal?
• Composition with identity function
• Composition with an inverse function
• Composition of two one-to-one functions is
  one-to-one
• Composition of two onto functions is onto

20
Cardinality

• Cardinality refers to the size of the set
• Finite and infinite sets
• Two sets have the same cardinality when there is
  bijective function associating them
• Cardinality is is reflexive, symmetric and
  transitive
• Countable sets set of all integers, set of even
  numbers, positive rationals (Cantor
  diagonalization)
• Set of real numbers between 0 and 1 has same
  cardinality as set of all reals
• Computability of functions