Lecture 37 21 127 Concepts of Math 12 05 2012 Administrivia Homework 10 is due tomorrow Extra Credit projects Due on Sunday at noon If you can turn it in on Friday in lecture that would be better just because I want you to be done with them so that you can start studying Homework Comment About the Pigeonhole Problem HW 10 4 Pretend there is someone who proved that d 14 for instance notice 14 12 The point of the second part is that you can guarantee his proof is wrong without even reading it Probability Probability Spaces Definition Let S be a finite set We will call S the probability space Elements of S are called outcomes and subsets of S are called events We want a probability function that satisfies the following properties Pr P S R That is the probabilty function should be a well defined function that takes in an event a subset of S and outputs a real number A S 0 Pr A 1 Pr S 1 A B S If A B then Pr A B Pr A Pr B That s it These are the most basic properties we want a probability function to have These are our axioms From them we can prove many facts including A S Pr A 1 Pr A Pr Pr S 1 1 0 A B S Pr A B Pr A Pr B Pr A B By induction If A1 A2 An are pairwise disjoint then n X Pr A1 A2 An Pr Ak k 1 Example Rolling 2 Dice Consider rolling two fair distinguishable dice one die is colored white and the other is black In this case our probability space S is the set of all ordered pairs of numbers from 1 to 6 That is S 6 6 Each outcome is equally likely since the dice are fair so we want our probability function on this space to weight every outcome the same Since there are 36 possible outcomes S 36 then we want the 1 That is probability of every outcome to be 36 x y S Pr x y 1 1 36 Notice that we take the probability of the set that contains that one outcome this is because the probability function is defined on P S the events In general what this means is that the probability of any event only depends on how many outcomes it contains that is X S Pr X X 36 This is because S is what we call a uniform probability space It is uniform in the sense that all outcomes are equally likely Many of the probability spaces we will work with will be uniform When this is the case you can use the following fact In a finite uniform probability space the probability of an event only depends on the number of outcomes in that event Now returning to the example at hand What is the probability that a roll of these two dice avoids doubles Let X be the event that the roll is not a double We can find Pr X by finding 1 Pr X that is let us first find the probability that a roll is a 6 double and subtract that from 1 This probability is 36 16 since there are 6 ways to roll a double Thus Pr X 1 Pr X 1 5 1 6 6 What is the probability that a roll of these two dice makes the sum of the two numbers a multiple of 4 Let Y be the event that this happens To have the sum be a multiple of 4 that sum must be 4 or 8 or 12 We can enumerate the possibilities for each of these 3 cases 1 3 3 1 2 2 2 6 6 2 3 5 5 3 4 4 6 6 We see that there are 9 possibilities so 9 1 36 4 What is the probability that a roll of these two dice yields a sum that is a multiple of four while also avoiding doubles This question is looking for the probability of the event that both X and Y happen simultaneously That is we want Pr Y 6 1 Pr X Y Pr 1 3 3 1 2 6 6 2 3 5 5 3 36 6 Notice in particular that this is not Pr X Pr Y Pr X Y 6 Pr X Pr Y 1 6 1 4 15 36 because 5 12 That is X Y 6 What is the probability that the number on the white die is strictly less than the number on the black die Let Z be the event that this occurs We need to find Z to be able to find Pr Z We can find Z by observing that whenever the black die shows n there are n 1 options for what the white die can show Thus we have Z 6 X n 1 0 1 2 3 4 5 15 n 1 and so Pr Z 15 5 36 12 2 What is the probability that the white die shows a 4 Let W be the event that this occurs We see that W 6 because there are exactly 6 options for the black die s number after setting the white die s number to be 4 Thus 1 6 Pr W 36 6 Now here is an interesting question What is the probability that the white die s number is strictly less than the black die s number given the information that the white die s number is a 4 Intuitively we want to say 31 because only 2 of the 6 options for the black die s number are larger than 4 But why does this work How does this come from our formal definitions of probability and probability spaces This is what the next subsection addresses We will continue working with this example of rolling two dice to illustrate the concept of conditional probability Conditional Probability We often wonder about how the probability of some event is or isn t affected by some outside knowledge we have gained That is what if we actually knew that some event already occurred How does this change perhaps the probability of some other event also happening This is usually phrased in terms of some given knowledge In the dice example we are working with we want to know the probability that the white die s number is strictly less than the black die s number given the information that the white die s number is a 4 In terms of the events we defined this is asking What is the probability of Z given the knowledge that W occurs We can answer this by essentially redefining our probability space to be W instead of S Since we are given that the white die shows a 4 we really only need to consider the outcomes where that event definitely happens Any outcomes in S W are not possible so we throw them away for the time being Then we just need to see how likely it is that …