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Chapter 7 Lesson 7.5 Random Variables and Probability Distributions 7.5: Binomial and Geometric Distributions Special Distributions Two Discrete Distributions: Binomial and Geometric One Continuous Distribution: Normal Distributions Properties of a Binomial Experiment 1. There are a fixed number of trials 2. Each trial results in one of two mutually We use n to denote the fixed exclusive outcomes. (success/failure) number of trials. 3. Outcomes of different trials are independent 4. The probability that a trial results in success is constant. The binomial random variable x is defined as x = the number of successes when a binomial experiment is performed Are these binomial distributions? 1) Toss a coin 10 times and count the number of heads Yes 2) Deal 10 cards from a shuffled deck and count the number of red cards No, probability does not remain constant 3) The number of tickets sold to children under 12 at a movie theater in a one hour period No, no fixed number Binomial Probability Formula: Let n = number of independent trials in a binomial experiment p = constant probability that any trial results in a success Where: n! x n x P (x ) p (1 p ) x ! (n x )! n n ! Technology, n C xsuch as calculators and x software, x ! (n x )!will also statistical perform this calculation. Let’s record the gender of the next 5 newborns at Huntington Memorial Hospital and see how many girls we get. Is this a binomial experiment? What is the probability of Yes, if the births“success”? were not multiple births (twins, etc). Define the random variable of interest. x = the number of females born out of the next 5 births What are the possible values of x? x 0 1 2 3 4 5 Newborns Continued . . . What is the probability that exactly 2 girls will be born out of the next 5 births? P (x 2) 5 C 2 0.5 0.5 .3125 2 3 What is the probability that less than 2 girls will be born out of the next 5 births? P (x 2) p (0) p (1) 5 C 0 .5 .5 5 C 1 .5 .5 0 .1875 5 1 4 Newborns Continued . . . Let’s construct the discrete probability distribution table for this binomial random variable: x 0 1 2 3 p(x) .03125 .15625 .3125 .3125 4 5 .15625 .03125 is the multiplying WhatNotice is the that meanthis number ofsame girls as born in the next five births? n×p Since this is a +discrete mx = 0(.03125) + 1(.15625) 2(.3125) + distribution, could use: 3(.3125) + 4(.15625)we + 5(.03125) =2.5 mx xp Formulas for mean and standard deviation of a binomial distribution mx np x np 1 p Newborns Continued . . . How many girls would you expect in the next five births at a particular hospital? mx np 5(.5) 2.5 What is the standard deviation of the number of girls born in the next five births? x np (1 p ) 5(.5)(.5) 1.118 The Binomial Model (cont.) Binomial Probability Model n = number of trials p = probability of success q = 1 – p = probability of failure X = # of successes in n trials x n-x P(X = x) = nCx p q m = np Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley s = npq Independence One of the important requirements is that the trials be independent. When we don’t have an infinite population and we are sampling without replacement, the trials are not independent. But, there is a rule that allows us to pretend we have independent trials: The 10% condition: If the trials are not independent, it is still okay to proceed as long as the sample is smaller than 10% of the population. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 17- 13 Newborns Revisited . . . Suppose we were not interested in the number of females born out of the next five births, but which birth would result in the first female being born? How is this question different from a binomial distribution? Properties of Geometric Distributions: • There are two possible outcomes: a success or failure So what are the • Each trial is independent of the others possible values of x • The probability of success is constant for all trials. To infinity How far will this go? A geometric random variable x is defined as x = the number of trials UNTIL the FIRST success is observed ( including the success). x 1 2 3 4 . . . Probability Formula for the Geometric Distribution Let p = constant probability that any trial results in a success x 1 p (x ) (1 p ) Where x = 1, 2, 3, … p Suppose that 40% of students who drive to campus at your school or university carry jumper cables. Your car has a dead battery and you don’t have jumper cables, so you decide to stop students as they are headed to the parking lot and ask them whether they have a pair of jumper cables. Let x = the number of students stopped before finding one with a pair of jumper cables Is this a geometric distribution? Yes Jumper Cables Continued . . . Let x = the number of students stopped before finding one with a pair of jumper cables p = .4 What is the probability that third student stopped will be the first student to have jumper cables? P(x = 3) = (.6)2(.4) = .144 What is the probability that at most three student are stopped before finding one with jumper cables? P(x < 3) = P(1) + P(2) + P(3) = (.6)0(.4) + (.6)1(.4) + (.6)2(.4) = .784 The Geometric Model (cont.) Geometric probability model for Bernoulli trials: p = probability of success q = 1 – p = probability of failure X = # of trials until the first success occurs x-1 P(X = x) = q p 1 m= p Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley s= q p2 Homework • Pg.439: #7.44, 47, 52-54, 56, 58, 60, 61