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Probability Models The Bernoulli Family

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What is a Bernoulli trial? 3 characteristics: -two possibilities (yes/no, true/false, success/failure) -constant probability of success -all events are independent or your sample is less than 10% of the population After slide Page 1.2 – 1.3

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Bernoulli Trials Geometric: Geom(p) Mean & stdev How long till first success Binomial: Bion(n, p) Mean & Stdev r successes given n trials The Bernoulli Family

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Give 2 examples of a Bernoulli trial? After slide Page 1.5 3 characteristics: -two possibilities (yes/no, true/false, success/failure) -constant probability of success -all events are independent

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Two types of Bernoulli trials. Page 1.6 after Geometric: (infinite series) applies to waiting time situations – counting the number of trials to achieve our first success. How many components until the first success? Binomial: (finite series) count the number of successes we get in a given number of trials. P(4 successes given 10 trials)

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Skittles! Page 1.7 The manufacturer says that there are 20% of the red ones that I like best. I jiggle my machine and one slips out. -Two outcomes: red or other -Probability of red is 20% -Getting one candy does not effect the next candy.

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Skittles! Page 1.8 What is the probability that the first red candy is the 4 th candy gotten out of the machine? This is a Geometric Probability Model Looking for the FIRST success of a Bernoulli trial They are consecutive, one after the other so we are multiplying. P(red c ) P(red c ) P(red c ) P(red)

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Skittles! What is the probability that the first red candy is the 4 th candy selected? They are consecutive, one after the other so we are multiplying. P(red c ) P(red c ) P(red c ) P(red) (0.8)(0.8)(0.8)(0.2) = ? Or P(1 st red is the n th ) = q n-1 p Where q is the P(failure)

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Skittles! Page 1.9 E(X=x), Stdev 1.10 -20 skittles likely? What is the probability that the first red candy is the 4 th candy selected? Lets create a probability model for getting a red skittle out of the machine Find the E(X=x) & stdev of the model Geom(p) on average, how many need to come out before you get a red one? Create a probability model.

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Skittles! Page 1.11 What is the probability that the first red candy is one of the first 3 that comes out? {could be the 1 st or 2 nd or 3 rd } This is a Binomial Model – 1 red in 3 trials We are adding probabilities here. P(1 st ) + P(2 nd ) + P(3 rd ) (0.8) 0 (0.2)+(0.8) 1 (0.2)+(0.8) 2 (0.2) = ? 1 st 2 nd 3 rd

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Skittles! What is the probability that out of 3 candies, one will be red? 3 x P(1 red) = why times 3? 3 x (0.8) 2 (0.2)= How many ways can 1 red skittle show up? 3 C 1

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Skittles! What is the probability that out of 3 candies, one will be red?.2.8 (0.8) 2 (0.2)

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Skittles 3 page 1.12 – which model, how many ways, prob. of 4 of 12 What is the probability that out of 12 candies, four will be red? {just try to do a tree out of this one! I dare you}

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Skittles 3 page 1.13 – values of n, r, p, q What is the probability that out of 20 candies, 5 will be red?

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Bernoulli Trials Geometric: Geom(p) Mean & stdev How long till first success Binomial: Bion(n, p) Mean & Stdev r successes given n trials The Bernoulli Family

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Donating Blood Page 1.14 1 only The percent of the population that donates blood and is O-positive is 6%. The Utah Red Cross anticipates the need for at least 1600 units of O-negative blood this year. It estimates that it will collect blood from 32,000 donors. How great is the risk that the Red Cross will fall short of meeting its need? Is this a geometric or binomial? (1) Write a formula to find P(exactly 1600 units) Can the calculator do this? What can we do to solve this if the calculator cant do it?

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NMNM

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Find the P(X>1600) Page 1.14 do 2-4

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Conditions for a Bernoulli Trial 1.only 2 possibilities 2.fixed probability for the possibilities 3.independent or less than 10% of the population

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