Statistics%20270%20-%20Lecture%205

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Statistics 270 - Lecture 5
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• Last class measures of spread and box-plots
• Last Day - Began Chapter 2 on probability.
  Section 2.1
• These Notes more Chapter 2Section 2.2 and 2.3
• Assignment 2 2.8, 2.12, 2.18, 2.24, 2.30, 2.36,
  2.40
• Due Friday, January 27
• Suggested problems 2.26, 2.28, 2.39

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Probability

• Probability of an event is the long-term
  proportion of times the event would occur if the
  experiment is repeated many times
• Read page 59-60 on Interpreting probability

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Probability

• Probability of event, A is denoted P(A)
• Axioms of Probability
• For any event, A,
• P(S) 1
• If A1, A2, , Ak are mutually exclusive events,
• These imply that

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Discrete Uniform Distribution

• Sample space has k possible outcomes
  Se1,e2,,ek
• Each outcome is equally likely
• P(ei)
• If A is a collection of distinct outcomes from S,
  P(A)

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Example

• A coin is tossed 1 time
• S
• Probability of observing a heads or tails is

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Example

• A coin is tossed 2 times
• S
• What is the probability of getting either two
  heads or two tails?
• What is the probability of getting either one
  heads or two heads?

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Example

• Inherited characteristics are transmitted from
  one generation to the next by genes
• Genes occur in pairs and offspring receive one
  from each parent
• Experiment was conducted to verify this idea
• Pure red flower crossed with a pure white flower
  gives
• Two of these hybrids are crossed. Outcomes
• Probability of each outcome

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Note

• Sometimes, not all outcomes are equally likely
  (e.g., fixed die)
• Recall, probability of an event is long-term
  proportion of times the event occurs when the
  experiment is performed repeatedly
• NOTE Probability refers to experiments or
  processes, not individuals

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Probability Rules

• Have looked at computing probability for events
• How to compute probability for multiple events?
• Example 65 of SFU Business School Professors
  read the Wall Street Journal, 55 read the
  Vancouver Sun and 45 read both. A randomly
  selected Professor is asked what newspaper they
  read. What is the probability the Professor
  reads one of the 2 papers?

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• Addition Rules
• If two events are mutually exclusive
• Complement Rule

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• Example 65 of SFU Business School Professors
  read the Wall Street Journal, 55 read the
  Vancouver Sun and 45 read both. A randomly
  selected Professor is asked what newspaper they
  read. What is the probability the Professor
  reads one of the 2 papers?

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Counting and Combinatorics

• 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
• These methods are often useful for computing
  probabilites

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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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Product Rule

• 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 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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Tree Diagram

• Can help visualize the possible outcomes
• Constructed by listing the posbilites for E1 and
  connecting these separately to each possiblility
  for E2, and so on

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Example

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

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Example - Permuatation

• 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 k at a time
• Sometimes denoted Pk,n
• Can be viewed as number of ways to select k
  things from n objects where the order matters

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Permutations

• The number of ordered sequences of k objects
  taken from a set of n distinct objects (I.e.,
  number of permutations) is
• Pk,nn(n-1) (n-k1)

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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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Combinations

• If one is not concerned with the order in which
  things occur, then a set of k 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 k objects
  selected from n objects is
• n choose k
• Note n!n(n-1)(n-2)1
• Note 0!1
• Can be viewed as number of ways to select mthings
  taken k 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 Conservatives, 3 Liberals and 2 NDPs
• How many committees have a member from each
  group?
• What is the probability that there is a member
  from each group on the committee?