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Probability Models The Bernoulli Family

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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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What is a Bernoulli trial? Q1: do not use wikipedia! 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

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Give 2 examples of a Bernoulli trial? Q2: 2 examples 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. Q3: give an example of each! Geometric: applies to waiting time situations – counting the number of trials to achieve our first success. How many components until the first success? Binomial: count the number of successes we get in a given number of trials. P(4 successes given 10 trials)

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Skittles! Q4: is this a Bernoulli trial? Check the assumptions. 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! Q5: is this a geometric or binomial? 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! What is the probability that the first red candy is the 4 th candy selected? This is a Geometric Probability Model Looking for the FIRST success of a Bernoulli trial Find the expected value of getting a red skittle; on average, how many do you need to select to get a red one? Create a probability model.

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Skittles! Q6 & Q7: what is the formula for the mean and stdev of a Geom(p) model What is the probability that the first red candy is the 4 th candy selected? Find the stdev of the model Geom(p) And the stdev

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Skittles! Q8: is this a geometric or binomial? 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 } 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) = ?

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Skittles! Q9: how many different ways can you get 1 red out of 3 total? What is the probability that out of 3 candies, one will be red?.2.8

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Skittles! What is the probability that out of 3 candies, one will be red? 3 x P(1 red) 3 x (0.8) 2 (0.2)= ?

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Skittles! What is the probability that out of 3 candies, one will be red? This is a Binom(n, p) model Its mean or E(X) = np Its stdev or = npq

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Skittles 3 Q10: is this a geometric or binomial? 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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Q11: how many ways can you get 4 reds out of 12 skittles

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Skittles 3 Q12a-e: identify n, r, p, q, and n-r What is the probability that out of 12 candies, four 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 -2 outcomes -Constant P(x) -independent The Bernoulli Family

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6.2 BINOMIAL PROBABILITIES. Features Fixed number of trials (n) Trials are independent and repeated under identical conditions Each trial has.

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