Statistics Chapter 8

Size: px

Start display at page:

Download "Statistics Chapter 8"

Robyn Manning
5 years ago
Views:

1 Statistics Chapter 8 Binomial & Geometric Distributions Time: weeks Activity: A Gaggle of Girls The Ferrells have 3 children: Jennifer, Jessica, and Jaclyn. If we assume that a couple is equally likely to have a girl or a boy, then how unusual is it for a family to have 3 children who are all girls? We have encountered problems like this in an earlier chapter. This time we ll use simulations. If success = girl, and failure = boy, then p(success) = 0.5. We will define the random variable X as the number of girls. Then we want to simulate families with 3 children. Our goal is to determine the longterm relative frequency of a family with 3 girls, that is, P(X=3). 1. Using a random number generator, let even digits represent girl and odd digits represent boy. Select 3 digits at a time. Each 3 digits will constitute one trial. Use tally marks to record the results: 3 girls Not 3 girls Do at least 40 trials. Then combine your results with those of other students in the class to obtain at least 200 trials. Calculate the relative frequency of the event {3 girls}. 2. For programming experts, write a calculator program to carry out the process described above. Allow the user to specify the number of trials, and have the calculator report the relative frequency of {3 girls} as a decimal number. 3. Determine the total number of outcomes for this experiment. List the outcomes in the sample space. Then complete the probability distribution table for the random variable X = number of girls. X Do the results of your simulations come close to the theoretical value for P(X=3)? P(X)

2 Introduction: We frequently encounter situations where there are 2 distinct outcomes. Binomial & geometric distributions are very closely related with only one major difference. That difference is: Identifying Binomial Experiment: Binomial Distributions: The distribution notation of X is noted as B(n,p) (or sometimes p is shown as ) Is it a binomial distribution? # of heads in 3 tosses # of rolls til you get a 6 # of free throws made in 5 tries # of aces in 4 draws without replacement # of aces in 4 draws with replacement Working toward the formulas: If Amy is a 70% free throw shooter and she takes 3 shots, what are the possible outcomes and their probabilities? P(SSS) = (.7)(.7)(.7) = P(SSF) = (.7)(.7)(.3) = P(SFS) = (.7)(.3)(.7) = etc. Let X = the number of shots she makes in 3 tries. X B(3,.7) 2

3 P(3 of 3) =? P(2 of 3) =? P(1 of 3) =? P(0 of 3) =? Formula: If X B(n,p) and k = # of successes: P(X=k) = n (p) k (1-p) n-k for k = 0, 1, 2,...n k Suppose you rolled a die 4 times and let x = number of sixes. Find the probability distribution of X and the probability that you get 2 or more sixes. X = # of sixes X B(4,1/6) Using the calculator: Pdf: probability distribution function Cdf: cumulative distribution function Suppose that a certain basketball player makes 20% of her 3- point shots and that she takes 4 3-point shots in a particular game. What would be the probability that she make at most 2 shots? 3

4 If I roll a die 30 times, what is the probability I get between 2 and 4 sixes (inclusive)? Let x = # of sixes then X B(30,1/6) Binomial Mean & Standard Deviation If X has the binomial distribution with the number of observations n and probability of success p, The mean is x = np The standard deviation is σx = np(1-p) If 20% of BHS students are Hispanic, find the mean and standard deviation of the number of Hispanics in a random sample of size 10. Use the TI-83 and the rules for discrete random variables. Suppose that 85% of all customers at a gas station choose 87 octane gas. If you observe the next 50 customers, a. How many customers do you expect to choose 87 octane gas? b. What is the standard deviation of the number of customers that choose 87 octane gas? c. What is the probability that at least 80% of customers choose 87 octane gas? 4 d. What is the probability that the number of customers who choose 87 octane gas is within 2 standard deviations of the mean?

5 Geometric Distribution In a geometric distribution, we are interested in how long it will take to achieve one successful trial unlike a binomial distribution which is interested in the number of successes in a fixed number of trials. X which is the number of trials until the first success is observed (including the success trial) Properties of a Geometric Experiment: Geometric probability distribution: X G(p) and is calculated by the formula: P(X=n) = (1-p) n-1 p Suppose a couple has a 25% chance of having a blond child: a. What is the probability they will have a blond child on the first try? X P(X) p (1-p)p (1-p) 2 p (1-p) 3 p (1-p) 4 p (1-p) 5 p (1-p) 6 p b. On their second try? Third try? Fourth try? nth try? c. How many trials can it take? 5

6 Suppose that 10% of the shoppers at the mall are senior citizens. If you were to observe people walk by a certain point in the mall, (remember to define variable and distribution) a. What is the probability that you won t see a senior citizen until the 20th observation? b. What is the probability that it will take at least 5 observations to get your first senior citizen? Mean & Standard Deviation of Geometric Random Variable If X G(p) then: Mean (or expected value) is = 1/p Variance of X is (1 p) / p 2 The probability that Eric gets a hit is.300. How many at bats, on average, does he need to get his first hit? What is the probability that it takes him less than average? What is the probability that it takes more than n trials to see the first success? P(X>n) = (1-p) n 6

What is the probability that it takes more

Using the Calculator: Suppose we roll a die

histogram. 1. Enter numbers 1 to 10 in L 1.

reasonable window X[1,11] and Y[-.05,.2] 3.

7 What is the probability that it takes more than 6 rolls to observe a 3? Using the Calculator: Suppose we roll a die until a 3 is observed. Let s have the calculator calculate the probability distribution table and plot a histogram. 1. Enter numbers 1 to 10 in L 1. Enter the probabilities into L 2. (Enter geometpdf(1/6,l 1 ) into L 2 ). 2. Specify dimensions for viewing window a reasonable window X[1,11] and Y[-.05,.2] 3. Statplot 1 with L 1 and L 2 4. Install cdf in L Statplot L 1 and L 3 specify viewing window: X[0,11], and [-.3,1] press graph 7

8 Green M&M s Suppose that 10% of regular M&M s are green. Randomly select M&M s until you get a green M&M. Let X = # of M&M s needed to get one green M&M.... X G(.1) Chapter 8 problems: # 1, 2, 3, 6, 7, 10, 12, 17, 19, 22, 26, 37, 39, 44, 45, 46 8

Chapter 8: The Binomial and Geometric Distributions

Chapter 8: The Binomial and Geometric Distributions 8.1 Binomial Distributions 8.2 Geometric Distributions 1 Let me begin with an example My best friends from Kent School had three daughters. What is the