Statistics 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 A 1, A 2, …, A k are mutually exclusive events, These imply that

Discrete Uniform Distribution Sample space has k possible outcomes S={e 1,e 2,…,e k } Each outcome is equally likely P(e i )= 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