Probability Expected Payoffs and Expected Utility In thinking about mixed strategies we will need to make use of probabilities probabilities We will therefore review the basic rules of probability and then derive the notion of expected value We will also develop the notion of expected utility as an alternative to expected payoffs Probabilistic analysis arises when we face uncertainty In situations where events are uncertain uncertain a probability measures the likelihood that a particular event or set of events occurs e g The probability that a roll of a die comes up 6 The probability that two randomly chosen cards add up to 21 Bl Blackjack kj k Sample Space or Universe Let S denote a set collection or listing g of all possible states of the environment known as the sample space or universe a typical state is denoted as s For example S s1 s2 success failure or low high price S s1 s2 sn 1 sn list of units sold or offers received S 0 stock price or salary offer continuous positive set space Events An event is a collection of those states s that result in the occurrence of the event An event can be that state s occurs or that multiple states occur or that one of several states occurs there are other possibilities Event A is a subset of S denoted as A S Event E t A occurs if th the ttrue state t t s iis an element of the set A written as s A Venn Diagrams Illustrates the sample space and events S A1 A2 S is the sample space and A1 and A2 are events within S Event A1 does not occur Denoted A1c Complement of A1 Event A1 or A2 occurs Denoted A1 A2 For probability use Addition Rules Event A1 and A2 both occur denoted A1 A2 For probability use Multiplication Rules Probability To each uncertain event A or set of events e g A1 or A2 we would like to assign weights which measure the likelihood or importance of the events in a proportionate manner Let P Ai be the probability of Ai We further assume that Ai S all i P Ai 1 all i P Ai 0 Addition Rules The probability of event A or event B P A B If the events do not overlap i e the events are disjoint subsets of S so that A B then the probability of A or B is simply the sum of the two probabilities P A B P A P B If the events overlap are not disjoint so that A B use the modified addition rule P AUB P A P B P A B Example Using the Addition Rule Suppose you throw two dice There are 6x6 36 possible ways in which both can land Event A What is the probability that both dice show the same number A 1 1 2 2 3 3 4 4 5 5 6 6 so P A 6 36 Event B What is the probability that the two die add up to eight B 2 6 3 5 4 4 5 3 6 2 so P B 5 36 Event C What is the probability that A or B happens i e P A B First note that A B 4 4 so P A B 1 36 P A B P A P B P A B 6 36 5 36 1 36 10 36 5 18 Multiplication Rules The probability of event A and event B P A B Multiplication M ltiplication rule r le applies if A and B are independent events A and B are independent events if P A does not depend on whether B occurs or not and P B does not o depend depe d oon w whether e e A occu occurss oor not o P A B P A P B P AB Conditional probability for non independent events The probability of A given that B has occurred is P A B P AB P B E Examples l U Using i M Multiplication lti li ti Rules R l An unbiased coin is flipped pp 5 times What is the probability of the sequence TTTTT P T 5 5 independent flips so 5x 5x 5x 5x 5 03125 Suppose a card is drawn from a standard 52 card deck Let B be the event the card is a queen P B 4 52 E A Conditional C d l on Event E B what h is i P B 4 52 Event A B the probability that the card is the Queen of Hearts First note that P AB P A P AB P Ah B 1 52 Probability the Card is the Queen of Hearts P A B P AB P B 1 52 4 52 1 4 Bayes Rule Suppose events are A B and not B i e Bc Then Bayes rule can be stated as P B A P A B P B P A B P B P A B c P B c Example Suppose a drug test is 95 effective the test will ill be b positive i i on a drug d user 95 off the h time and will be negative on a non drug user 95 of the time Assume 5 of the population are drug users Suppose an individual tests positive What is the probability he is a drug user B Bayes Rule R l Example E l Let A be the event that the individual tests positive Let B be the event individual is a drug p y event that user Let Bc be the complementary the individual is not a drug user Find P B A P A B 95 95 P A Bc 05 05 P B P B 05 05 P Bc 95 95 P A B P A B P B P B A P A B P B P A B c P B c 95 05 50 95 05 05 95 Montyy Hall s 3 Door Problem There are three closed doors Behind one of the doors is a brand new sports car Behind each of the other two doors is a smelly goat You can t see the car or smell the goats You win the prize behind the door you choose The Th sequence off play l off the th game iis as ffollows ll You choose a door and announce your choice The host Monty Hall who knows where the car is always selects one of the two doors that you did not choose which he knows has a goat behind it Monty then asks if you want to switch your choice to the unopened door that you did not choose Should you switch 1 2 3 You Should Always Switch Let Ci be the event car is behind door i and let G be the event Monty chooses a door with a goat behind it Suppose without loss of generality the contestant chooses door 1 Then Monty shows a goat behind door number 3 1 and so P G C1 1 1 According …