Binomial and Geometric Distributions Section 3.2 & 3.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH Department of Mathematics University of Houston February 11, 2016 Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department of Mathematics February 11, 2016 University1 of/ Hous 23
Outline 1 Beginning Questions 2 Binomial Distribution 3 Geometric Distribution Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department of Mathematics February 11, 2016 University2 of/ Hous 23
Popper Set Up Fill in all of the proper bubbles. Use a #2 pencil. This is popper number 04. Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department of Mathematics February 11, 2016 University3 of/ Hous 23
Popper Questions Wallen Accounting Services specializes in tax preparation for individual tax returns. Data collected from past records reveals that 9% of the returns prepared by Wallen have been selected for audit by the Internal Revenue Service (IRS). 1. What is the probability that a customer of Wallen will be selected for audit? a. 0.09 b. 0.91 c. 1 d. 0 2. Today, Wallen has six new customers. Assume the chances of these six customers being audited are independent. Let X = the number of customers out of the six will be audited. What is the probability that none of the six will be selected for audit? That is P(1st is not audited "and" 2nd is not audited "and" 3rd is not audited "and" 4th is not audited "and" 5th is not audited "and" 6th is not audited). a. 0 b. 1 c. 0.568 d. 0.432 Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department of Mathematics February 11, 2016 University4 of/ Hous 23
Audit Example Let X be the number out of 6 customers that will be audited. The possible values of X are X = {0, 1, 2, 3, 4, 5, 6} What is the probability that exactly one out of the 6 customers will be audited? athy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department of Mathematics February 11, 2016 University5 of/ Hous 23
Audit Example From this example we can note a couple of things. 1. We are only looking at six customers, n = 6. 2. We are assuming the chances of these six customers being audited are independent. 3. For each customer, they will either be selected for audit or not be selected for audit. 4. We have the same probability of a person being selected for audit, p = 0.09 for each customer. This type of setting occurs quite frequently that we can put a mathematical model to these types of probability. Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department of Mathematics February 11, 2016 University6 of/ Hous 23
Binomial Setting The previous example falls into a Binomial Setting which follows these 4 rules. 1. There are a fixed number n of observations. 2. The n observations are all independent. That is, knowing the result of one observation tells you nothing about the other observations. 3. Each observation falls into one of just two categories, we call success and failure. 4. The probability of a success p is the same for each observation. Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department of Mathematics February 11, 2016 University7 of/ Hous 23
Popper Questions Do these scenarios have a binomial setting? 3. Rolling a die 50 times. a. Yes b. No 4. Rolling a die 50 times and finding the number of times that 5 occurs. a. Yes b. No 5. Determining the number of heads up out of three flips. a. Yes b. No Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department of Mathematics February 11, 2016 University8 of/ Hous 23
Binomial Probability Distribution The distribution of the count X of successes in the Binomial setting has a Binomial probability distribution. Where the parameters for a binomial probability distribution is: n the number of observations p is the probability of a success on any one observation The possible values of X are the whole numbers from 0 to n. As an abbreviation we say, X B(n, p). Binomial probabilities are calculated with the following formula: P(X = k) = n C k p k (1 p) n k athy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department of Mathematics February 11, 2016 University9 of/ Hous 23
Steps to fining the probability 1. Make sure the assumptions are met for the Binomial setting. 2. Describe the success. 3. Determine p, the probability of success. 4. Determine n, the number of trials. 5. Put the probability question in terms of X. That is, P(X = x) or P(X x) or P(X x) or P(X < x) or P(X > X). 6. Determine the probability. We can do this through the formula,in our calculator or R. Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 10 of/ Hous 23
Using the Binomial Formula With the example of Wallen customers being selected for audit, n = 6 and p = 0.09. What is the probability that exactly one of the six will be selected for audit? 1. We have independence. 2. Success is being audited. 3. p = 0.09 4. n = 6 5. We want P(X = 1). We have met the 4 conditions of the binomial setting so we can use the formula: P(X = k) = n C k p k (1 p) (n k). P(X = 1) = 6 C 1 (0.09) 1 (0.91) 5 = 0.337 Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 11 of/ Hous 23
Using R or TI-83\84 To find probabilities in R. To find P(X =k) use dbinom(k,n,p). From previous example P(X = 1), k = 1, n = 6, p = 0.09 > dbinom(1,6,0.09) [1] 0.3369774 To find P(X k) use pbinom(k,n,p). To find probabilities in TI-83\ 84 use the DISTR button this will give you a list of probability distributions. To find P(X = k) use binompdf(n,p,k). To find P(X k) use binomcdf(n,p,k). Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 12 of/ Hous 23
Example #2 A fair coin is flipped 30 times. 1. What is the probability that the coin comes up heads exactly 12 times? P(X = 12), n = 30, p = 0.5 > dbinom(12,30,0.5) [1] 0.08055309 2. What is the probability that the coin comes up heads less than 12 times? P(X < 12) = P(X 11) > pbinom(11,30,0.5) [1] 0.1002442 3. What is the probability that the coin comes up heads more than 12 times? P(X > 12) = 1 P(X 12) > 1-pbinom(12,30,0.5) [1] 0.8192027 Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 13 of/ Hous 23
Mean and Variance of a Binomial Distribution If a count X has the Binomial distribution with number of observations n and probability of success p, the mean and variance of X are µ X = E[X] = np σ 2 X = Var[X] = np(1 p) Then the standard deviation is the square root of the variance. Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 14 of/ Hous 23
From text 3.2 # 17 Suppose it is known that 80% of the people exposed to the flu virus will contract the flu. Out of a family of five exposed to the virus, what is the probability that: 1. No one will contract the flu? 2. All will contract the flu? 3. Exactly two will get the flu? 4. At least two will get the flu? Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 15 of/ Hous 23
Example Continued Suppose it is known that 80% of the people exposed to the flu virus will contract the flu. Suppose we have a family of five that were exposed to the flu. 1. Let X = number of family members contacting the flu. Create the probability distribution table of X. 2. Find the mean and variance of this distribution. Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 16 of/ Hous 23
Popper Question Fifty percent of married couples paid for their honeymoon themselves. You randomly select 10 married couples and ask each if they paid for their honeymoon themselves. 6. What is the probability that exactly 5 out of the 10 paid for the honeymoon themselves. a. 0.5 b. 0.00488 c. 0.0009 d. 0.2461 7. What is the expected value, µ X of the number of married couples that paid for their honeymoon themselves out of 10 couples asked? a. 5 b. 10 c. 0 d. 5.5 8. What is the standard deviation, σ X, of the number of married couples that paid for their honeymoon themselves out of the 10 couples asked? a. 5 b. 2.5 c. 1.5811 d. 6.25 Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 17 of/ Hous 23
Geometric Distribution The geometric distribution is the distribution produced by the random variable X defined to count the number of trials needed to obtain the first success. Examples: Flipping a coin until you get a head, Rolling a die until you get a 5. athy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 18 of/ Hous 23
Conditions A random variable X is geometric if the following conditions are met: 1. Each observation falls into one of just two categories, "success" or "failure." 2. The probability of success is the same for each observation. 3. The variable of interest is the number of trials required to obtain the first success. Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 19 of/ Hous 23
Formula for Geometric Distribution The probability that the first success occurs on the n th trial is: P(X = n) = (1 p) n 1 p Where p is the probability of success. The probability that it takes more than n trials to see the first success is: P(X > n) = (1 p) n R commands:p(x = n) = dgeom(n-1,p) and P(X > n) = 1 pgeom(n 1, p) TI-83\84: P(X = n) = geometpdf(p,n) and P(X > n) = 1 geomtcdf(p, n) Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 20 of/ Hous 23
Mean and Variance of a Geometric Distribution If a count X has the Geometric distribution with probability of success p, the mean and variance of X are µ x = E[X] = 1 p σ 2 x = Var[X] = 1 p p 2 The standard deviation is the square root of the variance. Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 21 of/ Hous 23
From Text section 3.3 #8 A quarterback completes 44% of his passes. We want to observe this quarterback during one game to see how many passes he makes before completing one pass. 1. What is the probability that the quarterback throws 3 incomplete passes bfore he has a completion? 2. How many passes can the quarterback expect to throw before he completes a pass? 3. Determine the probability that it takes more than 5 attempts before he completes a pass. 4. What is the probability that he attempts more than 7 passes before he completes one? Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 22 of/ Hous 23
From Text Section 3.3 #14 Newsweek in 1989 reported that 60% of young children have blood lead levels that could impair their neurological development. Assuming a random sample from the population of all school children at risk, find: 1. The probability that at least 5 children out of 10 in a sample taken from a school may have a blood lead level that may impair development. 2. The probability you will need to test 10 children before finding a child with a blood lead level that may impair development. 3. The probability you will need to test no more than 10 children before finding a child with a blood lead level that may impair development. Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th Section 2:30 pm3.2-5:15 & 3.3 pm 620 PGH (Department offebruary Mathematics 11, 2016 University 23 of/ Hous 23