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Bernoulli trial • Bernoulli Trial ~ a trial with 2 outcomes success or failure (arbitrary names) • p denotes P (success) • P (failure)= 1-p A Binomial Experiment • A binomial experiment is a series of Bernoulli trials done to determine • X ~ the number of successes. • X is a discrete random variable. • n denotes the number of Bernoulli trials. • The Trials are independent • discrete means that the set of values that the RV can take on is countable or countably infinite. (think a set of individual values) Probability Distribution n k nk p ( X k ) p (1 p ) k Remember that X is a random variable being the number of successes after n trials. p is the probability of success. NOTE: this is another common way to write combinations. n nCk k Your Book’s Equation • The Equation is written this way in the text book. It is identical, they use q to represent 1-p and express the combination as nCx as they use x to represent the particular value of the random variable X being considered. p ( x) n C x p q x You are welcome to use either form. n x Expected value n E(X ) x P(X i x i ) np i 1 The short cut to the expected value of X in a binomial experiment is np Example: 4 dice are rolled, how many 1’s are likely to appear? • This is a binomial experiment • Each die roll is a Bernoulli Trial with success~ 1 fail ~ 2 through 6 • X ~ the number of 1’s • X={0,1,2,3,4} • Probability Distribution • P(X=k)=(nCk)pk(1-p)n-k • • • • • P(X=0)=4C0(5/6)4=0.48 P(X=1)=4C1(1/6)(5/6)3=.386 P(X=2)=4C2(1/6)2(5/6)2=.116 P(X=3)=4C3(1/6)3(5/6)1=.015 P(X=4)=4C4(1/6)4=.00077 Expected Value E(X) n E(X ) x P(X i xi ) i 1 = x1P(X=x1)+ x2P(X=x2)+ x3P(X=x3)+…+ xnP(X=xn) • E(X)=0*0.48+1*0.386+2*.116+3*.015 +4*.00077 = 0.666 But Remember E(X)=np • 4 trials, each with a probability p=1/6 of success • E(X)=np=4*1/6=0.667 • (this answer is more accurate as we did not round until the end) A company is producing brake callipers. • Probability of a defect is 1.2% • If 150 brake callipers are produced, what is the probability that no more than 2 are defective? Analyze the Event • If X represents the number of defective callipers, the event that no more than 2 are defective is the event that X≤2. • If X ≤ 2 then X=0 or X=1 or X=2 • P(X ≤2)=P(0) +P(1)+P(2) P(X ≤2)=P(0) +P(1)+P(2) • p=0.012 150 p ( X 0 ) 0 150 p ( X 1) 1 1-p=0.988 n=150 0 150 0 . 012 ( 0 . 988 ) 16 . 35 % 1 149 0 . 012 ( 0 . 988 ) 29 . 79 % 150 p ( X 2 ) 2 2 148 0 . 012 ( 0 . 988 ) 26 . 95 % • P(X ≤2)=P(0) +P(1)+P(2) • =16.35% + 29.79% + 26.95% • =73.09% Re Cap • When ever you recognize that a series of Bernoulli trial are being done to determine the number of successes this is a binomial experiment. • The outcomes probabilities are distributed n according to : p ( X k ) k p (1 p ) k nk • The expected value E(X) = np if X is the result of a binomial experiment. Practice • Page 385 1 to 12, 14