10/8/09 1 FPP 13-14 Probability Probability  What statisticians hang their hat on  Provides a formal framework from which uncertainty can be quantified  Why study probability in an intro stat course?  Lay foundations for statistical inference.  Train your brain to think in a way that it is not hardwired to do  Its quite enjoyable and relaxing10/8/09 2 Types of probability  What exactly is probability?  There are any number of notions of probability, indicating that probability isn’t a thing but a concept  We can spend a semester philosophizing about probability if you are interested I can direct you to some books.  An unexhausted list  Laplacian probability  Hypothetical limiting relative frequency probability  Nomic probability  Fiducial probability  Epistemic probability  In this class we will focus on two of these. Heads up  Mathematical notation will become a little more prevalent here. You’ll need to put forth effort wrapping your brain around it.10/8/09 3 Limiting relative frequency  Most folks call this the frequentist approach 1. Operations: observation, measurement, or selection that can at least hypothetical ly be repeated an infinite number of times 2. Sample space: set of possible outcomes of an operation 3. Events: subsets of elements in the sample space  Elements of the sample space (basic outcomes) are equally likely  Calculation 1. Let S denote the sample space, E ⊂ S denote an event, and |A| denote the size of any set A 2. P r(E) ≡ |E|/|S |  Upshot  Percentage of times an event occurs in repeated realizations of random processes Epistemic probability  Often times called “subjective” probability  This term is a bit loaded as it can be argued that “objective” probability doesn’t really exist  Here probability is degree of belief in likelihood of event  Belief is updated or modified in the light of observed information10/8/09 4 Probability  Why consider two probabilities  Each allows different approaches to incorporating probability in an anslysis  Each one leads to different types of inference statements.  Is one preferable to the other?  This really depends on who you ask.  There have been (heated) discussions on the appropriateness of both Frequency probability  We focus first on how to use frequency probability in an analysis and will cover epistemic probability later  Simple motivating example  There are 3 red balls and 9 white balls in a hat  Pick one ball at random out of the hat  Once picked the ball is not replaced  Then pick another ball at random out of the hat10/8/09 5 Shorthand for probability  Define R1 = pick a red ball on the 1st try  Define R2 = pick a red ball on the 2nd try  Define W1 = pick a white ball on the 1st try  Define W2 = pick a white ball on the 2nd try  Probability of picking a red ball on the 1st try is  Pr(R1) =  Probability of picking two red balls in two picks without replacing 1st ball is  Pr(R1 and R2) = Marginal and joint probability  Probability of a single event is called marginal probability  Example: Pr(R1)  Probability of intersection of two events (both events happening) is called a joint probability  Example: Pr(R1 and R2)10/8/09 6 Conditional probability  Say we pick a red ball on the 1st try. The chance we pick a red ball on the 2nd try equals 2/11.  Probability that an event given another event occurs is called conditional probability  Shorthand: Pr(R2|R1) = 2/11  “Probability that R2 occurs given that R1 occurs.” Relating these probabilities  Pr(R1 and R2) = Pr(R1)Pr(R2|R1)  6/132 = 3/12(2/11)  Joint prob. = marginal prob. times conditional prob.  This is always true10/8/09 7 Independent events  Replace 1st ball before picking the 2nd . Then  Pr(R1) = 3/12  Pr(R2 | R1) = Pr(R2) = 3/12  R1 and R2 are called independent events: The occurrence of R1 does not affect the probability of R2. Independent events  When events are independent calculating joint probabilities is fairly easy  Let events A, B, C, … etc. be independent  Pr(A and B and C and … etc. ) = Pr(A)Pr(B)Pr(C)Pr(etc.)  To get joint probabilities you can simply multiply the marginal probabilities  Why does this work?10/8/09 8 Dependent events  Notice that when sampling with out replacement then  Pr(R2|R1) = 2/11 ≠ 3/12 = Pr(R2)  When the conditional prob. is not equal to the marginal prob. then the events are said to be dependent.  The occurrence of R1 affects the probability of R2  Here R1 and R2 are dependent events Dependent events  When events are dependent joint probabilities are harder to compute  Let A, B, C, …, etc. be dependent events  Pr(A and B and C and … etc.) = Pr(A|B,C,etc.)Pr(B|C, etc.)Pr(C|etc.)Pr(etc.)  To get joint probabilities, you multiply all the conditional probabilities10/8/09 9 Genetic inheritance  One of two genes, A or a, inherited from each parent  Say Pr(A) = 0.5 for each parent  Genes inherited independently from parents  Pr(A from mom and A from dad) = Pr(A)Pr(A) = 0.25 Independence in sports  Baseball announcers sometimes say, “The batter has not gotten a base hit in the last four times he’s batted. He’s due for a hit now”  What is this statement assuming?10/8/09 10 “or” rule  Pr(R1 or R2 ) = Pr(R1) + Pr(R2) – Pr(R1 and R2) = Pr(R1) + Pr(R2) – Pr(R1)Pr(R2) Why? = 3/9 + 3/9 – (3/9)(3/9) “or” rule  If events are disjoint (i.e. they cannot happen simultaneously) then we can split “or” probabilities into sums of individual probabilities  These are also referred to as “mutually exclusive” events  Pr(W1 or R1) = Pr(W1) + Pr(R1) - Pr(W1 and R1) = Pr(W1) + Pr(R1) - 0  I know that this example is quite unenlighting10/8/09 11 Better “or” example embedded into total law of probability  Law of total probability  P(A) = P(A and B) + P(A and not B)  P(Brown eyes)=P(Brown eyes and Male)+P(Brown eyes and Female)  R2 occurs in two ways 1. Red