Counting

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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?

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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.
• (Cant be processed simultaneously).
• Here is an example of a possible schedule
• Question What is the number of all possible
  schedules?

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Situations where counting techniques are used

• Consider the following nested loop
• for i1 to 5
• for j1 to 6
• Statement 1
• Statement 2 .
• next j
• next i
• Question How many times the statements in the
  inner loop will be executed?

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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 ).

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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.

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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

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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 ij7
• (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
  
• n(S) 62 36 , n(E) 6.
• Thus, chances of winning P(E) 6/36 1/6 .

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Number of Elements in a List

• If m and n are integers and m n ,
• then there are n-m1 integers
• from m to n inclusive.
• Example
• a) How many elements are there in the array
  A12, A13, , A75, A76 ?
• b) What is the probability
• that a randomly chosen element of the array
  has a subscript which is divisible by 7 ?

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Number of Elements in a List

• Example (cont.)
• Solution a) 76 12 1 65 .
• b) Sample space
• S Ai 12 i 76 .
• Event that the index is divisible by 7
• E Ai 12 i 76 and 7i .
• n(S) 65 from part (a).
• 1472, 2173, , 70710 .
• Thus, n(E) 10-21 9 .
• Hence the probability that the index is
  divisible by 7
• P(E) n(E) / n(S) 9 / 65 .14