1 Counting

2 Situations where counting techniques are used  You toss a pair of dice in a casino game. You win if the numbers showing face up have a sum of 7.  Question: What are your chances of winning the game?

3 Situations where counting techniques are used  To satisfy a certain degree requirement, you are supposed to take 3 courses from the following group of courses: CS300, CS301, CS302, CS304, CS305, CS306, CS307, CS308.  Question: In how many different ways the requirement can be satisfied?

4 Situations where counting techniques are used  There are 4 jobs that should be processed on the same machine. (Can’t be processed simultaneously). Here is an example of a possible schedule:  Question: What is the number of all possible schedules? Job 3Job 1Job 4Job 2

5 Situations where counting techniques are used  Consider the following nested loop: for i:=1 to 5 for j:=1 to 6 [ Statement 1 ; Statement 2. ] next j next i  Question: How many times the statements in the inner loop will be executed?

Counting and Probability  Suppose we toss two coins.  Question. What are the chances of getting 0, 1, 2 heads?  The set of all possible outcomes: S = {(H,H), (H,T), (T,H), (T,T)} Event of getting exactly one head corresponds to the subset {(H,T), (T,H)}. Thus, chances of getting exactly one head is 2 / 4 =.5 ( which is the same as 50% ).

7 Random Processes, Sample Space and Events  A process is called random if  a set of different outcomes are possible;  one of the outcomes is sure to occur;  but it is impossible to predict with certainty which outcome that will be.  A sample space is the set of all possible outcomes of a random process or experiment.  An event is a subset of a sample space.

8 Probability  If S is a finite sample space (in which all outcomes are equally likely), E is an event in S, then the probability of E is  Notation: For any finite set A, n(A) denotes the number of elements in A. Then

Example on Probability  You toss a pair of dice in a casino game. You win if the numbers showing face up have a sum of 7.  Question: What are your chances of winning the game?  Solution. Sample Space: S = { (1,1), (1,2), …, (6,6) } = { (i,j) |  i, j  1,…,6 } The event that the sum is 7: E = { (i,j) | i, j  1,…,6 and i+j=7 } = { (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) } n(S) = 6 2 = 36, n(E) = 6. Thus, chances of winning = P(E) = 6/36 = 1/6.

Applying the dice example in Monopoly Game Your opponent’s token is in one of the squares His turn consists of rolling two dice and moving the token clockwise on the board the number of squares indicated by the sum of dice values When his token lands on a property that is owned by you, you collect rent It is more advantageous to have houses or hotels on your properties because rents are much higher than for unimproved properties You might build houses or hotels on your properties before your opponent rolls the dice Suppose you own most of the squares following (clockwise) your opponent’s token. In which square should you build houses or hotels?

11 Number of Elements in a List  If m and n are integers and m ≤ n, then there are n-m+1 integers from m to n inclusive.  Example: a) How many elements are there in the array A[12], A[13], …, A[75], A[76] ? b) What is the probability that a randomly chosen element of the array has a subscript which is divisible by 7 ?

12 Number of Elements in a List Example (cont.): Solution: a) 76 – = 65. b) Sample space: S = { A[i] | 12 ≤ i ≤ 76 }. Event that the index is divisible by 7: E = { A[i] | 12 ≤ i ≤ 76 and 7|i }. n(S) = 65 from part (a). 14=7∙2, 21=7∙3, …, 70=7∙10. Thus, n(E) = = 9. Hence the probability that the index is divisible by 7: P(E) = n(E) / n(S) = 9 / 65 ≈.14