Statistics 270 - Lecture 5

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Statistics 270 - Lecture 5. Last class: measures of spread and box-plots Last Day - Began Chapter 2 on probability. Section 2.1 These Notes – more Chapter 2…Section 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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### Statistics 270 - Lecture 5

Last class: measures of spread and box-plots
• Last Day - Began Chapter 2 on probability. Section 2.1
• These Notes – more Chapter 2…Section 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
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
Probability
• Probability of event, A is denoted P(A)
• Axioms of Probability:
• For any event, A,
• P(S) = 1
• If A1, A2, …, Akare mutually exclusive events,
• These imply that
Discrete Uniform Distribution
• Sample space has k possible outcomes S={e1,e2,…,ek}
• Each outcome is equally likely
• P(ei)=
• If A is a collection of distinct outcomes from S, P(A)=
Example
• A coin is tossed 1 time
• S=
• Probability of observing a heads or tails is
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?
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
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
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?
• If two events are mutually exclusive:
• Complement Rule
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?
Counting and Combinatorics
• In the equally likely case, computing probabilities involves counting the number of outcomes in an event
• This can be hard…really
• Combinatorics is a branch of mathematics which develops efficient counting methods
• These methods are often useful for computing probabilites
Combinatorics
• Consider the rhyme

As I was going to St. Ives

I met a man with seven wives

Kits, cats, sacks and wives

How many were going to St. Ives?

Example
• In three tosses of a coin, how many outcomes are there?
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 (n1n2n3 …nk )
• Example (St. Ives re-visited)
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?
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
Example
• In three tosses of a coin, how many outcomes are there?
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
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
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
Permutations
• The number of ordered sequences of k objects taken from a set of n distinct objects (I.e., number of permutations) is:
• Pk,n=n(n-1) … (n-k+1)
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
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?
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
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?
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?