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Discrete Mathematics Lecture 7. Harper Langston New York University. 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?

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Discrete MathematicsLecture 7

Harper Langston

New York University

poker problems
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?
combinations with repetition
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
algebra of combinations and pascal s triangle
Algebra of Combinations and Pascal’s Triangle
  • The number of r-combinations from a set of n elements equals the number of (n – r)-combinations from the same set.
  • Pascal’s triangle: C(n + 1, r) = C(n, r – 1) + C(n, r)
  • C(n,r) = C(n,n-r)
binomial formula
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
probability axioms
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)
conditional probability
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
conditional probability example
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?
conditional probability extended
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?
bayes theorem
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, then:P(Bk | A) = P(A | Bk) * P(Bk) .

P(A | B1)*P(B1) + … + P(A | Bn)*P(Bn)

Application: Medical Tests (false positives, etc.)

independent events
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.
generic functions
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
generic functions1
Generic Functions
  • Arrow diagrams for functions
  • Non-functions
  • Equality of functions:
    • f(x) = |x| and g(x) = sqrt(x2)
  • Identity function
  • Logarithmic function
one to one functions
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
onto functions
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
inverse functions
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
pigeonhole principle
Pigeonhole Principle
  • If n pigeons fly into m pigeonholes and n > 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?
pigeonhole principle1
Pigeonhole Principle
  • Generalized Pigeonhole Principle: For any function f : X  Y acting on finite sets, if n(X) > k * N(Y), then there exists some y from Y so that there are at least k + 1 distinct x’s so that f(x) = y
  • “If n pigeons fly into m pigeonholes, and, for some positive k, m >k*m, then at least one pigeonhole contains k+1 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.
composition of functions
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) = n^2. 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
  • 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