1. 461191 Discrete Math Lecture 6: Discrete Probability
San Ratanasanya
CS, KMUTNB

2. Today’s topics Basic of Counting Review Administrivia Probability
Baye’s Theorem
Expected Value and Variance

3. Basic Counting Tools Basic rules of counting
Sum rule: either task 1 or task 2 have to be done
Product rule: both task 1 and task 2 have to be done
Task 1 and Task 2 are independent
Inclusion and Exclusion Principle
Exclude one that count more than once or irrelevant count
Include one that miss
Pigeonhole Principle
If there are k nest for at least k+1 pigeons, then there must be at least one nest that has more than one pigeons

4. Example
How many strings of five ASCII character contain the character at least once? (ASCII has 128 characters)

5. Example
A computer company receives 350 applications from computer graduates for a job planning a line of new Web servers. Suppose that 220 of these people majored in CS, 147 majored in business, and 51 majored both in CS and business. How many of these applications majored in CS nor in business?
|A  B| = = 316
 U - |A  B| = 350 – 316 = 34

6. Example g pigeonhole
A drawer contains a dozen brown socks and a dozen black socks, all un matched. A man takes socks out at random in the dark.
How many socks must he take out to be sure that he has at least two socks of the same color?
How many socks must he take out to be sure that he has at least two black socks? 3 14

7. Binomial Coefficient THEOREM 1: The Binomial Theorem
Let x,y,n be positive integer
THEOREM 2: PASCAL’S IDENTITY
Let n and k be positive integer with n > k then
THEOREM 3: Let n be positive integer, then

8. Example How many terms are therein the expansion of (x + y)15?
What is the difference between the coefficient of the term x8y3 and x5y6?
215
|C(11, 8) – C(11, 5)| = |165 – 462| = 297

9. Permutations and Combinations
Ordered arrangement of a set of distinct objects.
An ordered arrangement of r elements of a set is r-permutation
Combinations
An unordered selection of r elements from the set is r-combination
P(n, r) = n!
(n - r)!
C(n, r) = P(n, r) = n!
r! r!(n – r)!

10. Example
A group contain n men and n women. How many ways are there to arrange these people in a row if the men and women alternate?
In how many ways can a set of five letters be selected from the English alphabet?
2(n!)2
C(26, 5) = 65780

11. Generalized Permutations and Combinations
Permutations and Combinations with repetition
Permutations with Indistinguishable Objects
Distributing Objects into Boxes (4 cases)
Distinguishable objects and distinguishable boxes
Distinguishable objects and indistinguishable boxes
Indistinguishable objects and distinguishable boxes
Indistinguishable objects and indistinguishable boxes

12. Example General PC How many string of six letter are there?
How many ways are there to select three unordered elements from a set with five elements when repetition is allowed?
266
C(5+3-1, 3) = C(7, 3) = 35

13. Generating Permutations and Combinations
Counting permutations or combinations can solve many problems
Counting is sometimes not enough. We need to generate either one of them to find the solution.

14. Example Generate PC
What is the next permutation in lexicographic order after ?
Count number of permutations = n! -1 = 6! – 1 = 719
Lexicogrphic: aj < aj+1. Therefore, the next number is
(Kakuro) Let S = {1, 2, …, 9}. What are the subset that has 3 elements of S and their sum is 15?
Count number of combinations = 9! / 3!6! = 84
Generate subset: {1, 5, 9}, {1, 6, 8}, {2, 4, 9}, {2, 5, 8} ,{2, 6, 7}, {3, 4, 8}, {3, 5, 7}, {4, 5, 6}

15. Administrivia Midterm Exam is in next 2 weeks. Please be prepared!
All assignments MUST be submitted before Midterm.

16. Discrete Probability and Probability Theory

17. Finite Probability Laplace’s definitions:
Experiment – a procedure that yields one of a given set of possible outcomes.
Sample space – the set of possible outcomes from the experiment.
Event – a subset of the sample space.
Definition 1:
If S is a finite sample space of equally likely outcomes,
and E is an event, that is , a subset of S, then
the probability of E is p(E) = |E|
|S|

18. The Probability of Combinations of Events
Theorem 1:
Let E be an event in a sample space S. The probability of
the event E, the complementary event of E, is given by
p(E) = 1 – p(E)
Theorem 2:
Let E1 and E2 be events in sample space S. Then
p(E1  E2) = p(E1) + p(E2) – p(E1  E2)

19. Example
There are many lotteries now that award enormous prizes to people who correctly choose a set of six numbers out of the first n positive integers, where n is usually between 30 and 60. What is the probability that a person picks the correct numbers out of 40?
What is the probability that a positive integer selected at random from the set of positive integers not exceeding 100 is divisible by either 2 or 5?
1 / C(40, 6) = 1 / =
– 0.1 = 0.6

20. Probabilistic Reasoning
The Monty Hall Three-door Puzzle
You are asked to select one of the 3 doors to win the price.
Once you select the door, the host, who knows what is behind each door, he opens one of the other two doors that he know is a losing door.
Then he ask you whether you would like to switch doors.
Should you change doors or keep your original selection?
You should change the door whenever you have chance!
The probability of wining will change from 1/3 to 2/3

21. Assigning Probabilities
Let S be the sample space of an experiment with a finite or countable number of outcomes. We assign a probaility p(s) to each outcome s such that these 2 conditions must be met
0  p(s)  1 for each s  S
The function p() is called a probability distribution

22. Definition 1:
Suppose that S is a set with n elements. The uniform distribution
assigns the probability 1/n to each element of S.
Definition 2:
The probability of the event E is the sum of the probabilities of
the outcomes in E. That is,
(Note that when E is an infinite set, is a convergent infinite series.)

23. Example
Suppose that a die is biased so that 3 appears twice as often as each number but that the other five outcomes are equally likely. What is the probability that an odd number appears when we roll this die?
Let E = {1, 3, 5}
p(1) = p(2) = p(4) = p(5) = p(6) = 1/7; p(3) = 2/7
Then p(E) = 4/7

24. Conditional Probability and Independence
Definition 3:
Let E and F be events with p(F) > 0. The conditional probability of
E given F, denoted by p(E | F), is defined as
p(E | F) = p(E  F)
p(F)
Definition 4:
The events E and F are independent if and only if p(E  F) = p(E)∙p(F)

25. Example
What is the conditional probability that a family with two kids has two boys, given they have at least one boy?
Are the events E, that a family with three kids has of both sexes, and F, that this family has at most one boy, independent?
Let E = {BB}, F = {BB, GB, BG} then E  F = {BB}
p(E|F) = (1/4)/(3/4) = 1/3
Yes. Because p(E  F) = p(E)p(F) = 3/8

26. Bernoulli Trials and the Binomial Distribution
An experiment with only 2 outcomes either Success or Failure is called Bernoulli Trials.
Let p and q be a probability of success and failure, respectively. Then, it follows that p + q = 1
Bernoulli Trails are mutually independent.
Theorem 2:
The probability of exactly k successes in n independent Bernoulli trials,
with probability of success p and probability of failure q = 1 – p, is
b(k; n, p) = C(n, k)pkqn-k
If we consider it as a function of k, we call this function the Binomial Distribution.

27. Random variable Definition 5:
A random variable is a function fro the sample space of an experiment
to the set of real numbers. That is, a random variable assigns
a real number to each possible outcome.
Definition 6:
The distribution of random variable X on a sample space S
is the set of pairs (r, p(X = r)) for all r  X(S), where p(X = r) is
the probability that X takes the value r. A distribution is usually
described by specifying p(X = r) fro each r  X(S).

28. Example
Suppose that a coin is flipped three times. Let X(t) be the random variable that equals the number of heads that appear when t is the outcome. Find X(t).
X(HHH) = 3, X(HHT) = X(HTH) = X(THH) = 2,
X(TTH) = X(THT) = X(HTT) = 1, X(TTT) = 0
Because each of the eight possible outcomes
Has probability of 1/8, the distribution of X(t) is given by
P(X=3) = 1/8, P(X=2)=3/8, P(X=1) = 3/8, P(X=0) = 1/8

29. More Examples
Birthday Problem. What is the minimum number of people who need to be in a room so that the probability that at least two of them have the same birthday is greater than 1/2?
Example
1- pn = … 367-n
n = 22

30. Monte Carlo Algorithms
Deterministic Algorithms
Always proceed in the same way whenever given the same input.
Probabilistic Algorithms
Make random choices at one or more step for its efficiency in a huge number of possible case.
Mote Carlo Algorithms
The probability that the algorithm give the correct answer increases as more test carried out.
Details in Section 6.2 page 411

31. The Probabilistic Method
is used as existence proof to proof results about set S.
See Theorem 4 for example
Theorem 3:
If the probability that an element of a set S does not have a
particular property is less than 1,
there exists an element in S with this property.

32. Baye’s Theorem

33. Bayes’ Theorem
The probability that a particular event occurs on the basis of partial evidence, i.e., it is like conditional probability
Theorem 1:
Suppose that E and F are events from a sample space S such that p(E)  0 and p(F)  0. Then
P(F | E) = p(E | F)p(F)
p(E | F)p(F) + p(E | F)p(F)
Theorem 2: Suppose that E is an event from a sample space S and that F1, F2, …, Fn are mutually exclusive events such that
Assume that p(E)  0 and p(Fi)  0 for i = 1, 2, .., n. Then

34. Example
We have 2 boxes. The first box contains two green balls and seven red balls; the second box contains four green balls and three red balls. Bob selects a ball by first choosing one of the two boxes at random. He then selects one of the balls in this box at random. If Bob has selected a red ball, what is the probability that he selected a ball from the first box?
Details in Section 6.3 on page 417

35. Example: Bayesian Spam Filter
Uses information about previously seen messages whether an incoming message is spam.
Look for occurrences of particular word in messages.
See details in Section 6.3 on page 421

36. Expected Value and Variance

37. Expected Value (of Random Variable)
is a weighted average of the values of a random variable.
provides central point for the distribution of values of the random variable.
Definition 1: The expected value (or expectation) of
the random variable X(s) on the sample space S is equal to
Theorem 1: If X is a random variable and p(X = r) is the probability
that X = r, so that p(X = r) = , then

38. Variance
Definition 4: Let X be a random variable on a sample space S. The variance of X, denoted by V(X), is
The standard deviation of X is defined to be
Theorem 6: If X is a random variable on sample space S, then
V(X) = E(X2) – E(X)2

39. Average-case Computational Complexity
See Example 8-9 on page

40. Homework Section 6.1 8, 20, 35, 38, 41 Section 6.2
18, 22, 36, 37, 39, 40
Section 6.3
5, 10, 15, 18, 23
Section 6.4
2, 7, 9, 10, 22, 23
Supplementary
2, 3, 15, 16, 21, 22