Class Note for STAT 528 at OSU 13
Class Note for STAT 528 at OSU 13
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Date Created: 02/06/15
Stat 528 Autumn 2008 Elly Kaizar Probability Reading Sections 41 42 45 o Chance experiments 0 Venn diagrams and events Complements and or Empty and disjoint events 0 Probabilities The complement rule Addition rule for disjoint events The general addition rule lndependence the multiplication rule gtllt Guaranteeing independence gtllt Independent versus disjoint Probability c We are interested in drawing some conclusions from a sample This involves some risk 0 We use probability to quantify our certainty about a vari able in the population given our measures of the variable in the sample Only applies if we take a random sample How do we Characterize random phenomenon Chance experiments and events o A chance experiment or a random experiment is an activity in which the result may change if the experiment is repeated many times i it depends on chance eg tossing a coin will it rain tomorrow assessing the potency of a drug treatment 0 lt arises because some natural phenomenon is at work we introduce randomness eg in an experimental or survey design 0 The sample space S is the set of all possible outcomes in the experiment 0 An event is an outcome or collection of outcomes of a chance experiment An event is a subset of the sample space We use letters A B C to denote events Battery example Suppose we take a random sample of six batteries from a produc tion line and record Whether or not each battery is defective a What is the sample space for this Chance experiment Example events A there are more than three defective batteries in the random sample B there are fewer than ve defective batteries in the random sample Viewing events via Venn diagrams Sample space S o The large rectangle denotes the sample space 0 Shapes and lled in regions Within the rectangle are the events A B and C in this case 0 An event with nothing in it is called the empty event Q Complement o For an event A the complement of A AC consists of all events in S that are not in A Sample space 8 AC Battery Example A there are more than three defective batteries in the random sample b Describe in words the event the complement of A And o The intersection of two events A and B A and B con sists of all events common to both A and B Sample space S Battery Example A there are more than three defective batteries in the random sample B there are fewer than ve defective batteries in the random sample C Describe in words the event A and B Disjoint o A and B are disjoint or mutually exclusive if they have no events in common A and B is the empty set Q Sample space S Battery Example A there are more than three defective batteries in the random sample B there are fewer than ve defective batteries in the random sample 1 Give an example of two disjoint events for this Chance exper iment Or o The union of two events A and B A or B consists of all events contained in A or B or both Sample space S Battery Example A there are more than three defective batteries in the random sample B there are fewer than ve defective batteries in the random sample e Describe in words the event A or B Assigning probabilities 0 We assign probabilities to events We let PA denote the probability of the event A o The rules axioms of probability Suppose we have a Chance experiment with sample space S Then 1 0 g PltAgt g 1 for any event A 2 135 1 10 How we determine probabilities c We repeat our chance experiment experiment many times Then PltAgt 7 number of times A occurred i number of times we repeat the experiment c We can often think of probability as the long run relative frequency 11 Examples 1 If we ip a coin the sample space is S H T Then 2 Consider rolling a six sided die The sample space is S 1 2 3 4 5 6 If the die is fair we have Pltroll a 1 Pltroll a 2 Pltroll a 3 Pltrolla4 Pltrolla5 Pltrolla6 Thus Pltget any number on the die 138 1 3 Based on Exercise 433 Suppose that the PlN for a certain automatic teller machine ATM consists of 3 digits How many possible Ple are there What is the sample space How would you calculate the probability that a PlN assigned at random has at least one zero 12 The complement rule o For any event A PM 1 PA 0 Ex For the six sided dice example consider the event A the die roll is greater than one What is the probability of AC What is the probability of A 13 Addition rule for disjoint events o For any two disjoint events A and B PltA or B PltAgt 133 think of the Venn diagraml 14 More dice games For the die example de ne three events A roll a 2 4 or 6 B roll a 1 2 or 3 and C roll a 5 a What is HA P03 and HG b Which pairs of events are disjoint C Calculate the probability of either of the events in b occur LL 77 ring or 15 The general addition rule o For any two events A and B PltA or B PltAgt 133 PltA and B 0 Think of the Venn diagram again 0 Dice EX cont Which pairs of events are not disjoint Calculate the probability of either of these events occurring 16 Independence 0 Events A and B are independent events if the probabil ity of either one occurring is not affected by the other event occurring 0 The multiplication rule for independent events PA and B PAPB 0 Implications if A is independent of B then 1 AC is independent of BC 2 A is independent of BC and 3 AC is independent of B o Disjoint events can not be independent 17 A forensic example ln forensic science the probability that any two people match with respect to a given characteristic hair color blood type etc is called a probability of match Suppose that the frequencies of blood phenotypes in the population are as follows A B AB 0 42 10 04 44 a What is the probability that two randomly chosen people both have blood type A 18 A forensic example cont b Repeat the calculation in parta for the three other blood types c Find the probability that two randomly chosen people have matching blood types Note a person can only have one phenotype 19 A forensic example cont d The probability that two people do not match for a given characteristic is called discriminating power What is the discriminating power for the comparison of two peoples blood types in part c 20 Guaranteeing independence 0 Can be hard to validate 0 Sometimes we can check if events are independent See next example 0 Other times we use intuition to decide if events are inde pen dent Independence is often a good approximation to reality Random sampling is often the key to ensuring inde pendence 21 EM radiation example All current carrying Wires produce elecotromagnetic radi ation including the electrical Wiring running into through and out of our homes High frequency EM is thought to be a cause of cancer the lower frequencies associated With household current are generally assumed to be harmless The following table sum marizes the probability distribution for cancer sufferers and their Wiring con guration in the Denver area Leukemia Lymphoma Other cancers High frequency Wiring 0242 0047 0079 Low frequency Wiring 0391 0098 a What is the missing probability labeled in the above table 22 EM radiation example c0nt b What is the probability of having high frequency Wiring among cancer sufferers in the Denver area 23 EM radiation example c0nt C Is the event Having Leukemia77 independent of the event Having high frequency Wiring Explain 24 Independence Intuition Another Dice Example S roll a 1 2 3 4 5 or 6 A roll a 1 2 3 or 4 B roll an odd number ie 1 3 or 5 0 General Intuition l roll a die but do not show you the result PltAgt Now l tell you that l rolled an odd number PltA knowing that B happened PAB If the probability does not change then the events are inde pendent This is the conditional probability way of writing the de nition of independence 25 Independence Intuition Another Dice Example cont o Graphical Intuition l roll a die but do not show you the result PltAgt 23 133 12 Think of probability as area A picture of A and B independence Where A is shaded pink and B is hashed blue 12 0 23 1 26
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