Download Statistics 400 - Lecture 2

Statistics 270 - Lecture 4
• Last class: measures of spread and box-plots
• Have completed Chapter 1
• Today - Chapter 2
Probability
• “There is a 75% chance of rain tomorrow”
• What does this mean?
Definitions
• Probability of an outcome is a numerical measure of the chance of
the outcome occurring
• A experiment is any action whose outcome is uncertain
• Sample space, S, is the collection of possible outcomes of an
experiment
• Event is a set of outcomes
• Event occurs when one of its outcomes occurs
Example
• A coin is tossed 1 time
• S=
• Describe event of getting 1 heads
• Event with one outcome is called:
Example
• A coin is tossed 2 times
• S=
• Describe event of getting 1 heads and 1 tails
• Event with more than one outcome is called:
Review of Sets
• The union of two events, A and B, is the event consisting of
outcomes that are in either A or B or both
• The Intersection of two events, A and B, is the event consisting of
all outcomes that are in both A and B
• The complement of an event A, denoted A’, is the set of all
outcomes in the sample space that are not in A
Visually
• Union
• Intersection
• Complement
• Two sets, A and B, are said to be mutually exclusive if they have no
events in common
• Visually
Example
• Bag of balls has 5 red and 5 green balls
• 3 are drawn at random
• S=
Example (continued)
• A is the event that at least 2 green are chosen
• A=
• B is the event that 3 green are chosen
• B=
Example (continued)
•
•
• A’
Probability
•
Probability of an event is the long-term proportion of times the
event would occur if the experiment is repeated many times
Probability
•
Probability of event, A is denoted P(A)
•
Axioms:
•
•
•
•
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' )