Math Review Lecture 5

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Math ReviewLecture 5

• Evans School of Public Affairs
• University of Washington

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Lecture 5 Summation

• You may encounter this notation for summation
• The Greek letter is capital sigma. The numbers
  above and below it are the values of the
  subscript that you should sum over.
• Example x11, x22, x33, x44. Find

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Lecture 5 Weighted Averages

• Use weighted averages to find the average of
  elements of a whole that are not evenly weighted.
  
• Example We want to know the fraction of students
  who are white in a school district containing two
  schools. School A has 1,000 students and is 40
  white, while School B has 500 students and is 70
  white. What fraction of students in the district
  are white? Hint the answer isnt 55.

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Lecture 5 Choices

• Multiplication Principle If you have n choices
  to make, with m1 ways to make choice 1, and for
  each of these, m2 ways to make choice 2, and so
  on, with mn ways to make choice n, the total
  number of possible choices is equal to
  m1m2..mn
• Example If there are 3 roads from town A to town
  B and 2 roads from town B to town C, in how many
  ways can someone travel from A to C by way of B?

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Lecture 5 Choices

• Example Suppose Andy has 7 pairs of shorts, 8
  shirts, and 5 different pairs of shoes. If he is
  willing to wear any combination, how many
  different shorts-shirt-shoe choices does he have?
  
• Example A combination lock can be set to open to
  any 3-letter sequence. How many sequences are
  possible? How many sequences are possible if no
  letter is repeated?

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Lecture 5 Factorials

• Factorials When multiplying combinations, you
  often need to calculate products such as
  54321, the product of the integers from 5 to
  1. If n is a positive integer, n! (n factorial)
  denotes the product of all the natural numbers
  from n down to 1.
• n! n(n-1)(n-2)...(3)(2)(1)
• n! n(n-1)!
• 0! 1

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Lecture 5 Permutations

• A permutation of a set of elements is an ordering
  of the elements. For instance, there are six
  permutations of the letters A, B, C
• ABC, ACB, BAC, BCA, CAB, CBA
• Order counts when determining the number of
  permutations of a set of elements. This means
  that the event ABC is distinct from CBA or any
  other ordering of the three letters.
• Example How many batting orders are possible for
  a 9-person baseball team?

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Lecture 5 Permutations

• Sometimes we want to order only some of the
  elements in a set, rather than all of them. 
• Example A teacher has 5 books and wants to
  display 3 of them side by side on her desk. How
  many arrangements of 3 books are possible?
• We say that the possible arrangements are
  permutations of 5 things taken 3 at a time, and
  we denote the number of such permutations by 5P3.
  In other words, 5P3 60.

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Lecture 5 Permutations

• More generally, an ordering of r elements from a
  set of n elements is called a permutation of n
  things taken r at a time, and the number of such
  permutations is denoted nPr.
• If (where r n) is the number of permutations of
  n elements taken r at a time, then

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Lecture 5 Permutations

• Example Early in 2004, 4 candidates sought the
  Democratic nomination for president. In how many
  ways could voters rank their first, second and
  third choices?
• 4P3 ?

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Lecture 5 Combinations

• Earlier, we saw that there are 60 ways a teacher
  can arrange 3 of 5 different books on a desk.
  That is, there are 60 permutations of 5 things
  taken 3 at a time. Suppose now that the teacher
  does not wish to arrange the books on her desk,
  but rather wishes to choose, at random, 3 of the
  5 books to give a book sale to raise money for
  her school. In how many ways can she do this?

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Lecture 5 Combinations

• 60 is the number of all possible arrangements of
  3 books chosen from 5 however, the following
  arrangements would all lead to the same set of 3
  books being given to the book sale
• mysterybiographytextbook
• biographytextbook--mystery
• mysterytextbookbiography
• textbookbiography--mystery
• biographymysterytextbook
• textbook-mysterybiography

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Lecture 5 Combinations

• This list shows 6 different arrangements of 3
  books, but only one subset of 3 books. A subset
  of items selected without regard to order is
  called a combination. The number of combinations
  of 5 things taken 3 at a time is written 5C3.
  Since they are subsets, combinations are not
  ordered.
• where r n
• can also be written as read n choose
  r

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Lecture 5 Combinations

• Example How many different sets of three books
  can the teacher choose?
• Example From a group of 10 students, a
  committee is to be chosen to meet with the dean.
  How many different 3-person committees are
  possible?

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Lecture 5 Probability

• Outcomes that are not necessarily certain can be
  described with the concept of probability.
• A sample space is a list of all possible outcomes
  that could occur.
• With a die you can roll a 1,2,3,4,5, or 6. With a
  coin, you can toss a head or tail.
• In this case, each possible outcome has an equal
  probability of occurring.

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Lecture 5 Probability

• You can also think of the probability of an event
  as the proportion of days or weeks in which the
  event has occurred.
• Example Suppose that it has rained during two
  out of the last ten Independence Day shows in
  Seattle. Without any other information, what is
  the probability of rain during the next July 4th?

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Lecture 5 Probability

• Write the probability of an event A as P(A).
• If event A happened m times out of n, then
  P(A)m/n.
• The probability of rolling a 2 on a 6-sided die
  is P(2)1/6.
• Example Whats the probability of drawing a card
  of the suit spades out of a standard deck?

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Lecture 5 Probability

• Rules for probability
• The probability of an impossible event is 0.
• The probability of a certain event is 1.
• Probability must be between 0 and 1, inclusive.

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Lecture 5 Probability

• Probability of multiple events
• The chance that either event A or event B will
  occur is
• P(A) P(B)
• This assumes A and B cant both occur.
• Example Whats the probability of rolling a 4 or
  a 6 on a fair die?

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Lecture 5 Probability

• Probability of multiple events
• The chance of event A and event B occurring is
• P(A) P(B)
• This assumes A and B are independent.
• Example With a pair of dice, find the
  probability of rolling a 4 on one and 3 on the
  other.

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Lecture 5 Probability

• Probability of an event not occurring
• If you know the probability of A, you can also
  calculate the probability that A wont occur.
• P(Not A) 1 - P(A)
• Because A either occurs or it doesnt.
• Example If there is a 40 chance of rain, whats
  the chance of staying dry?

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Lecture 5 Probability

• Remember Venn Diagrams?

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Lecture 5 Probability

• Use Venn Diagrams to visualize the probability of
  multiple events.
• The chance that either event A or event B will
  occur is
• P(A) P(B) P(AB)
• AB is the overlapping part of A and B. We
  subtract it once because it has already been
  counted twice (as part of A and as part of B).

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Lecture 5 Probability

• Example An NGO gives summer jobs to two
  students, hoping that at least one of the
  students will take a regular job with the
  organization after graduation. The probability
  that each student will work for the NGO is 0.3
  and the probability that they both will is 0.1.
  Whats the probability that at least one of them
  will take a regular job with the NGO?