Today

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Today

• Today
• Reading
• Read Chapter 1 by next Tuesday
• Suggested problems (not to be handed in) 1.1,
  1.2, 1.8, 1.10, 1.16, 1.20, 1.24, 1.28

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Assignment 1
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Combinatorics (Section 1.4)

• In the equally likely case, computing
  probabilities involves counting the number of
  outcomes in an event
• This can be hardreally
• Combinatorics is a branch of mathematics which
  develops efficient counting methods

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Combinatorics

• Consider the rhyme
• As I was going to St. Ives
• I met a man with seven wives
• Every wife had seven sacks
• Every sack had seven cats
• Every cat had seven kits
• Kits, cats, sacks and wives
• How many were going to St. Ives?
• Answer

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Example

• In three tosses of a coin, how many outcomes are
  there?

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Multiplication Principle

• Let an experiment E be comprised of smaller
  experiments E1,E2,,Ek, where Ei has ni outcomes
• The number of outcome sequences in E is (n1 n2 n3
  nk )
• Example (St. Ives re-visited)

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Example

• In three tosses of a coin, how many outcomes are
  there?

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Tree Diagram

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Example

• In a certain state, automobile license plates
  list three letters (A-Z) followed by four digits
  (0-9)
• How many possible license plates are there?

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Example

• Suppose have a standard deck of 52 playing cards
  (4 suits, with 13 cards per suit)
• Suppose you are going to draw 5 cards, one at a
  time, with replacement (with replacement means
  you look at the card and put it back in the deck)
• How many sequences can we observe

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Permutations

• In previous examples, the sample space for Ei
  does not depend on the outcome from the previous
  step or sub-experiment
• The multiplication principle applies only if the
  number of outcomes for Ei is the same for each
  outcome of Ei-1
• That is, the outcomes might change change
  depending on the previous step, but the number of
  outcomes remains the same

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Permutations

• When selecting object, one at a time, from a
  group of N objects, the number of possible
  sequences is
• The is called the number of permutations of N
  things taken n at a time
• Sometimes denoted NPn
• Can be viewed as number of ways to select N
  things taken n at a time where the order matters

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Example

• Suppose have a standard deck of 52 playing cards
  (4 suits, with 13 cards per suit)
• Suppose you are going to draw 5 cards, one at a
  time, without replacement
• How many permutations can we observe

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Counting Patterns

• Consider the word minimum
• How many permutations of the letters are there?
• How many distinguishable ways are there to to
  arrange these letters?

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Counting Patterns

• The number of distinct sequences of N objects
  where m1 are are of type 1, m2 are are of type 2,
  , mk are are of type k is
• Note N m1 m2 mk

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Counting Patterns

• Consider the word minimum
• How many permutations of the letters are there?
• How many distinguishable ways are there to to
  arrange these letters?

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Example

• In a certain state, automobile license plates
  list three letters (A-Z) followed by four digits
  (0-9)
• How many possible license plates are there with 7
  distinct characters?

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Combinations

• If one is not concerned with the order in which
  things occur, then a set of n objects from a set
  with N objects is called a combination
• Example
• Suppose have 6 people,3 of whom are to be
  selected at random for a committee
• The order in which they are selected is not
  important
• How many distinct committees are there?

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Combinations

• The number of distinct combinations of n objects
  selected from N objects is
• N choose n
• Note N!N(N-1)(N-2)1
• Note 0!1
• Can be viewed as number of ways to select N
  things taken n at a time where the order does not
  matter

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Combinations

• Example
• Suppose have 6 people, 3 of whom are to be
  selected at random for a committee
• The order in which they are selected is not
  important
• How many distinct committees are there?

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Example

• A committee of size three is to be selected from
  a group of 4 Democrats, 3 Independents and 2
  Republicans
• How many committees have a member from each
  group?
• What is the probability that there is a member
  from each group on the committee?