Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Health Behavior Statistical Methods HP 340L Lecture 7 Probability Chapter 6Announcements: Office hours for Spring 2017 Day Time Place 1) Farzana Choudhury Tuesday 2.40-3.10 pm DML quad 2) Soo In Bang Thursday after class ZHS-159 3) Carol P. Wei Thursday 2.30-3.00 pm DML quad We covered material in Kiess & Green Chapter 6  The normal distribution  The standard normal distribution • Using normal tables • z-scores Last Lecture We will continue covering material in Kiess & Green Chapter 6  Basic probability Today’s Lecture What is probability theory?  The branch of mathematics that studies chance/random phenomena, e.g., games of chance  The ‘language’ in which mathematical statistics is ‘written’  For a user of statistics only the rudiments of probability are needed  We all have an intuitive notion of probability (and that’s pretty much all we’ll need)  “Probability theory is nothing but common sense reduced to calculation” – Pierre-Simon Laplace Probability Theory Examples:  What is a the probability that a coin will land on heads when flipped?  What is the probability that a 6-sided die will land on a 3 when rolled?  If dealt one card from a standard well-shuffled 52 card deck, what is the probability it’s an ace?  Pick one candy from a bag with 10 strawberry, 5 pear, and 15 peach flavored candy. What is the probability you get a pear flavored candy? Probability Theory What is probability?  What does it mean that the probability of landing heads is ½?  What is the probability of each possible outcome, i.e., the probability function? Probability Theory What does it mean that the probability of landing heads is ½?  Frequency of occurrence provides an intuitive way to interpret probabilities  If we flip a coin a large number of times, we’ll get heads about half the times  If we (hypothetically) flipped it infinitely many times, we would get heads exactly half of the times  ‘On average’ get heads half the time Probability Theory Probability “ingredients”  Probability experiment: a situation involving chance/randomness with several possible outcomes.  Example: in a die roll, outcomes are:  Event: a subset of possible outcomes  Example: a die roll is even: or or Probability Theory Notation  Possible outcomes:  Probability of an outcome:  P( ) = 1/6  P( ) = 1/6  Probability of an event, A:  A = { or or }  P(A) = 1/2 Probability Theory Where do probabilities come from?  From theory, e.g., equally likely outcomes:  P(coin flip = Heads) = 1/2  P(Die roll = 3) = 1/6  From experience/data (empirical), e.g., infant mortality rate (probability an infant will die before reaching age 1).  World: 49.4/1,000 = 4.9%  US: 5.4/1,000 = 0.5%  Japan: 2.62/1,000 = 0.2% Probability Theory Case of equally likely outcomes  If n total number possible outcomes and all are equally likely, probability of each individual outcome is 1/n.  Examples: fair die roll, coin toss, roulette  Probability of an event A is:  Example: A = die roll is even: Number of outcomes classifiable as AP(A)=Total number of possible outcomes3P(A)=6Probability Theory Basic properties of probabilities  Always between 0 and 1: 0 ≤ P(A) ≤ 1  Events certain to occur have probability 1, e.g., die roll ≤ 6  Events certain not to occur have probability 0, e.g., die roll > 6  The sum of the probabilities of all outcomes is 1 Probability Theory The additive rule  Outcomes that cannot occur at the same time are called mutually exclusive  Example: A die roll can be either even or odd, but not both  A= die roll is even, B=die roll is odd, are mutually exclusive  If A and B are mutually exclusive, the probability that either event A or event B occurs is the sum of the probabilities: P(A or B) = P(A) + P(B) Probability Theory Additive rule examples  Are the following events mutually exclusive?  Roll a die:  A= die roll is 1, B=die roll is even  A= die roll is 1, B=die roll is odd  A= roll is < 4, B = roll is < 3  A= roll is < 4, B = roll is > 3  Pick a student at random from the class:  A = student is US born; B = student is foreign born  A = US born; B = born in California  A = likes tacos; B = vegan  A = majors in Health Promotion; B = likes Statistics Probability Theory Yes No No No Yes No No No More additive rule examples  Are the following events mutually exclusive?  A= die roll is 1, B=die roll is even: Probability Theory Yes No  P(A) = 1/6; P(B) = 3/6; P(A or B) = 4/6  P(A or B) = P(A) + P(B) ✔  A= die roll is 1, B=die roll is odd:  P(A) = 1/6; P(B) = 3/6; P(A or B) = 3/6  P(A or B) ≠ P(A) + P(B)Examples for you to do on your own Deal a single card from a well shuffled deck:  What is the probability of getting hearts?  What is the probability of getting a 3?  What is the probability of getting a 3 of hearts?  What is the probability of an ace or a queen?  What is the probability of an ace or a queen of hearts? Probability Theory Example of outcomes that are NOT equally likely  Prof picks a student from the class to answer question  Only students that are present can be selected  Perhaps more likely to select from the first rows Probability Theory Discrete vs. continuous outcomes  All examples so far have a finite set of outcomes  When the total number of possible outcomes are finite (or countable, e.g., 1, 2, 3 … all the way to +∞), we call the set of outcomes discrete  When outcomes can take on a continuum of values we call the outcomes continuous  Examples:  Maximum temperature on a given day  An individual’s reaction time to a stimulus  Lake depth at different times of the year Properties of Continuous DistributionsDiscrete Outcome Continuous Outcome Probability function for heights  Probability functions Properties of Continuous