week 11 What is Probability? Quantification of uncertainty. Mathematical model for things that occur randomly. Random – not haphazard, don’t know what will happen on any one experiment, but has a long run order. The concept of probability is necessary in work with physical biological or social mechanism that generate observation that can not be predicted with certainty. Example… The relative frequency of such ransom events with which they occur in a long series of trails is often remarkably stable. Events possessing this property are called random or stochastic events

week 12 Basic Combinatorics Multiplication Principle Suppose we are to make a series of decisions. Suppose there are c 1 choices for decision 1 and for each of these there are c 2 choices for decision 2 etc. Then the number of ways the series of decisions can be made is c 1 ·c 2 ·c 3 ···. Example 1: Suppose I need to choose an outfit for tomorrow and I have 2 pairs of jeans to choose from, 3 shirts and 2 pairs of shoes that matches with this shirts. Then I have 2·3·2 = 12 different outfits.

week 13 Example 2: The Cartesian product of sets A and B is the set of all pairs (a,b) where a A, b B. If A has 3 elements (a 1,a 2,a 3 ) and B has 2 elements (b 1,b 2 ), then their Cartesian product has 6 members; that is A B = {(a 1,b 1 ), (a 1,b 2 ), (a 2,b 1 ), (a 2,b 2 ), (a 3,b 1 ), (a 3,b 2 )}. Some more exercise: 1. We toss R different die, what is the total number of possible outcome? 2. How many different digit numbers can be composed of the digits 1-7 ? 3. A questioneer consists of 5 questions: Gender (f / m), Religion (Christian, Muslim, Jewish, Hindu, others), living arrangement (residence, shared apartment, family), speak French (yes / no), marital status. In how many possible ways this questioneer can be answered?

week 14 Permutation An order arrangement of n distinct objects is called a permutation. The number of ordered arrangements or permutation of n objects is n! = n · (n – 1) · (n – 2) · · ·1 (“n factorial”). By convention 0! = 1. The number of ordered arrangements or permutation of k subjects selected from n distinct objects is n · (n – 1) · (n – 2) · · · (n – k +1). It is also the number of ordered subsets of size k from a set of size n. Notation: Example: n = 3 and k = 2 The number of ordered arrangements of k subjects selected with replacement from n objects is n k.

week 15 Examples 1.How many 3 letter words can be composed from the English Alphabet s.t: (i) No limitation (ii) The words has 3 different letters. 2.How many birthday parties can 10 people have during a year s.t.: (i) No limitation (ii) Each birthday is on a different day people are getting into an elevator in a building that has 20 floors. (i) In how many ways they can get off ? (ii) In how many ways they can get off such that each person gets off on a different floor ? 4.We need to arrange 4 math books, 3 physics books and one statistic book on a shelf. (i) How many possible arrangements exists to do so? (ii) What is the probability that all the math books will be together?

week 16 Combinations The number of subsets of size k from a set of size n when the order does not matter is denoted by or (“ n choose k”). The number of unordered subsets of size k selected (without replacement) from n available objects is Important facts: Exercise: Prove the above.

week 17 Example We need to select 5 committee members form a class of 70 students. (i) How many possible samples exists? (ii) How many possible samples exists if the committee members all have different rules?

week 18 The Binomial Theorem For any numbers a, b and any positive integer n The terms are referred to as binomial coefficient.

week 19 Multinomial Coefficients The number of ways to partitioning n distinct objects into k distinct groups containing n 1, n 2,…,n k objects respectively, where each object appears in exactly one group and is It is called the multinomial coefficients because they occur in the expansion Where the sum is taken over all n i = 0,1,...,n such that

week 110 Examples 1.A small company gives bonuses to their employees at the end of the year. 15 employees are entitled to receive these bonuses of whom 7 employees will receive 100$ bonus, 3 will receive 1000$ bonus and the rest will receive 3000$ bonus. In how many possible ways these bonuses can be distributed? 2.We need to arrange 5 math books, 4 physics books and 2 statistic book on a shelf. (i) How many possible arrangements exists to do so? (ii) How many possible arrangements exists so that books of the same subjects will lie side by side?