Transcript Statistics 400 - Lecture 2

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
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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
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Probability of event, A is denoted P(A)
•
Axioms of Probability:
• For any event, A, P( A)  0
• P(S) = 1
• If A1, A2, …, Ak are mutually exclusive events,
•
These imply that 0  P( A)  1
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?
• Addition Rules:
P( A  B)  P( A)  P( B)  P( A  B)
P( A  B  C )  P( A)  P( B)  P(C )  P( A  B)  P( A  C )  P( B  C )  P( A  B  C )
• If two events are mutually exclusive:
P( A  B)  P( A)  P( B)
• Complement Rule
P( A)  1  P( A' )
• 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
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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?
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Answer:
Example
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In three tosses of a coin, how many outcomes are there?
Product Rule
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Let an experiment E be comprised of smaller experiments
E1,E2,…,Ek, where Ei has ni outcomes
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The number of outcome sequences in E is (n1 n2 n3 …nk )
•
Example (St. Ives re-visited)
Example
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In a certain state, automobile license plates list three letters (A-Z)
followed by four digits (0-9)
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How many possible license plates are there?
Tree Diagram
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Can help visualize the possible outcomes
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Constructed by listing the posbilites for E1 and connecting these
separately to each possiblility for E2, and so on
Example
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In three tosses of a coin, how many outcomes are there?
Example - Permuatation
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Suppose have a standard deck of 52 playing cards (4 suits, with
13 cards per suit)
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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
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In previous examples, the sample space for Ei does not depend
on the outcome from the previous step or sub-experiment
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The multiplication principle applies only if the number of
outcomes for Ei is the same for each outcome of Ei-1
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That is, the outcomes might change change depending on the
previous step, but the number of outcomes remains the same
Permutations
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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
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The number of ordered sequences of k objects taken from a set of
n distinct objects (I.e., number of permutations) is:
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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
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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
n!
  
 Ck ,n
 k  (n  k )!n!
•
•
•
•
“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?
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What is the probability that there is a member from each group
on the committee?