10/13/09 Lecture 12 1STOR 155 Introductory StatisticsLecture 12: Birthday Problem, Prisoner Dilemma, Random VariableThe UNIVERSITY of NORTH CAROLINAat CHAPEL HILL10/13/09 Lecture 12 2Review10/13/09 Lecture 12 310/13/09 Lecture 12 410/13/09 Lecture 12 510/13/09 Lecture 12 6Birthday 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 73 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 83 Prisoners’ DilemmaA'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 9Free 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 10• 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. Random Variables10/13/09 Lecture 12 11Two types of random variables• A discrete random variable has a finitenumber 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 12Discrete Random Variable10/13/09 Lecture 12 13Flip 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.• Solution1/164/166/164/161/1610/13/09 Lecture 12 14Probability Histogram10/13/09 Lecture 12 15Questions• 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 16Continuous Random Variable (spinner)10/13/09 Lecture 12 17Continuous Random Variable• A continuous random variable X takes all possible values in an interval.– Not countable• The probability distribution of a continuous r.v. Xis described by a density curve. – What is a density curve?10/13/09 Lecture 12 18Ex: 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 19Continuous 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 20Continuous 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 21Women 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
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