AP Statistics: Geometric and Binomial Scenarios Objective: To understand similarities and differences between geometric and binomial scenarios and to solve problems related to these scenarios. Everything you need to know about Geometric and Binomial Models in 23 minutes: https://www.youtube.com/watch?v=vsmzwa3gp-k Porinchak, D. (2016, Dec. 14). APStatistics Binomial and Geometric Models --Youtube Common characteristics of Geometric and Binomial Scenarios (Bernoulli Trials) S I P Geometric Model Only concerned with. What is the probability I will my first success on x trial? P(X=x) = What is the expected number of trials I need to conduct before reaching my first success? µ = Standard Deviation: σ = Binomial Model How many successes will we get in n trials. We need. What is the probability we will get x successes in n trials? P(X=x) = where ( nn xx ) is the number of ways of finding x successes in n trials. What is the expected number of success that I will get in n trials? µ = Standard Deviation: σ = AP Stats Geometric and Binomial Scenarios 1
Quick Review of Combinations: The number of combinations is calculated when you have some number of trials (or options) and you want to choose x number out of the those options. Some ways of noting combinations: ( nn xx ) = nn! xx! nn xx! ncr -> read as n choose r For example, I have 5 books on my book shelf. I want to take 3 of those books with me on vacation. How many combinations of books can I take? ( 5 3 ) = 5! 3! 5 3! = 10 Ten combinations of 3 books out of 5 5C3 Chart it out: B1, B2, B3 B1, B2, B4 B1, B2, B5 B1, B3, B4 B1, B3, B5 B1, B4, B5 B2, B3, B4 B2, B3, B5 B2, B4, B5 B3, B4, B5 AP Stats Geometric and Binomial Scenarios 2
Geometric Example: Donating Blood People with O-negative blood are called universal donors because O- negative blood can be given to anyone else, regardless of the recipient s blood type. Only about 6% or the people have O-negative blood. Donor s line up at random for the blood drive. 1. Does this scenario meet the SIP criteria? Why or why not? Yes. S each person is either o-neg (success), or not (failure) I The blood type of one person does not affect the blood type of the next person in line. P the probability that any one person has o-neg blood is 6% 2. How many people do you expect to examine before you find someone who has O-negative blood? μ = 1 pp = 1 0.6 16.7 3. What is the probability that the first O-negative donor found is one of the first four people in line? P(X 4) = P(X=1) + P(X+2) + P(X=3) + P(X=4) = (0.06) + (0.94)(0.06) + 0.94 2 (0.06)+ 0.94 3 (0.06) 0.2193 AP Stats Geometric and Binomial Scenarios 3
Binomial Example: Donating Blood Same scenario. People with O-negative blood are called universal donors because O-negative blood can be given to anyone else, regardless of the recipient s blood type. Only about 6% or the people have O-negative blood. Donor s line up at random for the blood drive. 1. Suppose 20 donors come to the blood drive. What are the mean and standard deviation of the number of universal donors among them? This scenario still meets the SIP criteria and this scenario has a specific number of trials. We have n, and n=20. μ = np = (20)(0.06) =1.2, we expect 1.2 (people to have o-neg blood among n=20 people. σ= (npq) = (20)(0.06)(0.94) 1.06, so we may have between 0.14 people and 2.26 people with o-neg blood in n=20 people. 2. What is the probability that there are 2 or 3 universal donors? P(X = 2 or 3) = P(X=2) + P(X=3) =( 20 2 )(0.06)2 (0.94) 18 + ( 20 3 )(0.06)3 (0.94) 17 0.2246 + 0.0860 0.3106 AP Stats Geometric and Binomial Scenarios 4
Calculator tips Geometric models to find the probability of any individual outcome geometpdf(p,x) calculates finding the first success on the xth trial To find the sum of individual outcomes geometcdf(p,x) calculates finding the first success on or before the xth trial Binomial models to find the probability of any individual outcome binompdf(n,p,x) calculates the probability of finding exactly x number of successes in n trials To find the sum of individual outcomes binomcdf(n,p,x) calculates total probability of finding x or fewer number of successes in n trials Note: pdf is short for probability density function cdf is short for cumulative density function AP Stats Geometric and Binomial Scenarios 5