The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 12 Birthday Problem Prisoner Dilemma Random Variable 10 13 09 Lecture 12 1 Review 10 13 09 Lecture 12 2 10 13 09 Lecture 12 3 10 13 09 Lecture 12 4 10 13 09 Lecture 12 5 Birthday Problem In a classroom of 45 people what is the probability that at least two people have the same birthday Event A at least two people have the same birthday out of the 45 people AC every person has a different birthday out of the 45 people P A 1 P AC see the board 10 13 09 Lecture 12 6 3 Prisoners Dilemma Three prisoners A B and C are on a death row The governor decides to pardon one of the three and chooses at random the prisoner to pardon He informs the warden of his choice but requests that the name be kept secret for a few days The next day A tries to get the warden to tell him who had been pardoned The warden refuses A then asks which of B or C will be executed The warden thinks for a while then tells A that B is to be executed 10 13 09 Lecture 12 7 3 Prisoners Dilemma A s reaction Given that B will be executed then either A or C will be pardoned my chance of being pardoned has increased from 1 3 to 1 2 Q Is A correct Did the warden disclose any information to A Can A increase his chance of survival by swapping with C 10 13 09 Lecture 12 8 Free throws A TarHeel basketball player is a 80 free throw shooter Suppose he will shoot 20 free throws during each practice Which is more likely to make 5 out of 20 or 18 out of 20 How many free throws he makes on average during practice 10 13 09 Lecture 12 9 Random Variables Experiment A TarHeel basketball player shoots 20 free throws during his practice X number of hits A random variable is a variable whose value is a numerical outcome of a random experiment 10 13 09 Lecture 12 10 Two types of random variables A discrete random variable has a finite number of possible values X number of hits when trying 20 free throws Possible values for X 0 1 20 A continuous random variable takes values in an interval X the time it takes for a bulb to burn out Possible values are not countable 10 13 09 Lecture 12 11 Discrete Random Variable 10 13 09 Lecture 12 12 Flip a coin 4 times Find the probability distribution of the random variable describing the number of heads that turn up when a fair coin is flipped 4 times Solution 1 16 10 13 09 4 16 6 16 Lecture 12 4 16 1 16 13 Probability Histogram 10 13 09 Lecture 12 14 Questions What is the connection with histograms we talked about in Chapter 1 Are the two problems similar toss a coin 4 times and shoot 20 free throws Yes or no the free throw problem is equivalent to tossing a biased coin 20 times each with P H 0 8 10 13 09 Lecture 12 15 Continuous Random Variable spinner 10 13 09 Lecture 12 16 Continuous Random Variable A continuous random variable X takes all possible values in an interval Not countable The probability distribution of a continuous r v X is described by a density curve What is a density curve 10 13 09 Lecture 12 17 Ex Spinner continued P point to 1 4 0 why P greater than 5 8 1 5 8 3 8 P between 2 9 and 7 8 7 8 2 9 P falling in x x 1 4 1 4 for any x greater than 0 and less than 3 4 10 13 09 Lecture 12 18 Continuous Distribution The probability of any event is the area under the density curve and above the values of X that make up the event 10 13 09 Lecture 12 19 Continuous Distribution The probability model for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes In fact all continuous probability distributions assign probability 0 to every individual outcome The spinner Normal distributions are continuous probability distributions 10 13 09 Lecture 12 20 Women Height The height of American women aged 18 24 is approximately normally distributed with mean 64 3 inches and s d 2 4 inches Two women in the age group are randomly selected What is the probability that both of them are taller than 66 inches 10 13 09 Lecture 12 21
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