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(S) = 1

• If A1, A2, …, Ak are mutually exclusive events,

• These imply that 1)(0 AP

0)( AP

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:

• If two events are mutually exclusive:

• Complement Rule

)()()()( BAPBPAPBAP

)()()( BPAPBAP

)'(1)( APAP

)()()()()()()()( CBAPCBPCAPBAPCPBPAPCBAP