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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 13: Birthday Problem, Prisoner Dilemma, Random Variable 10/17/06 Lecture 13 1 Review 10/17/06 Lecture 13 2 10/17/06 Lecture 13 3 10/17/06 Lecture 13 4 10/17/06 Lecture 13 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) 10/17/06 Lecture 13 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. 10/17/06 Lecture 13 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 ? 10/17/06 Lecture 13 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? 10/17/06 Lecture 13 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. 10/17/06 Lecture 13 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. 10/17/06 Lecture 13 11 Discrete Random Variable 10/17/06 Lecture 13 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 10/17/06 4/16 6/16 Lecture 13 4/16 1/16 13 Probability Histogram 10/17/06 Lecture 13 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. 10/17/06 Lecture 13 15 Continuous Random Variable (spinner) 10/17/06 Lecture 13 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? 10/17/06 Lecture 13 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. 10/17/06 Lecture 13 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. 10/17/06 Lecture 13 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. 10/17/06 Lecture 13 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? 10/17/06 Lecture 13 21