3/1/11 Lecture 12 1 STOR 155 Introductory Statistics Lecture 12: Birthday Problem and Random Variables The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL3/1/11 Lecture 12 2 Review3/1/11 Lecture 12 33/1/11 Lecture 12 43/1/11 Lecture 12 53/1/11 Lecture 12 6 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)3/1/11 Lecture 12 7 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?3/1/11 Lecture 12 8 • 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 Variables3/1/11 Lecture 12 9 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.3/1/11 Lecture 12 10 Discrete Random Variable3/1/11 Lecture 12 11 Flip a fair 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 4/16 6/16 4/16 1/163/1/11 Lecture 12 12 Probability Histogram3/1/11 Lecture 12 13 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) = 0.8.3/1/11 Lecture 12 14 Continuous Random Variable (spinner)3/1/11 Lecture 12 15 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?3/1/11 Lecture 12 16 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 3/4.3/1/11 Lecture 12 17 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.3/1/11 Lecture 12 18 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.3/1/11 Lecture 12 19 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? (see more details on the board
View Full Document