#### Transcript 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' )