Probability ModelsThe Bernoulli Family

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

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

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)

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.

Skittles!Page 1.8

What is the probability that the first red candy is the 4th candy gotten out of the machine?

This is a Geometric Probability ModelLooking for the FIRST success of a

Bernoulli trialThey are consecutive, one after the other so we are multiplying . P(red c)

P(red c) P(red c) P(red)

Skittles!What is the probability that the first red candy is the 4th 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(1st red is the nth) = qn-1pWhere q is the P(failure)

Skittles!Page 1.9 E(X=x), Stdev 1.10 -20 skittles likely?

What is the probability that the first red candy is the 4th candy selected? Let’s create a probability model for getting a red skittle out of the machine

Find the E(X=x) & stdev of the model Geom(p)

2p

q

on average, how many need to come out before you get a red one? Create a probability model.

pXE

1)(

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 1st or 2nd or 3rd }

This is a Binomial Model – 1 red in 3 trials

We are adding probabilities here. P(1st) + P(2nd) + P(3rd )(0.8)0(0.2)+(0.8)1(0.2)+(0.8)2(0.

2) = ? 1st 2nd 3rd

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?

3C1

Skittles!What is the probability that out of 3 candies, one will be red?

.2

.2.2

.8

.8.8

(0.8)2(0.2)

(0.8)2(0.2)

(0.8)2(0.2)

Combinations: when the orderdoesn’t matter, just how many ways it can occur.

t want)don'you (what repeats

choices possible all

r

nC

rnr

nrn

)!(!

!

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}

Skittles 3page 1.13 – values of n, r, p, q

What is the probability that out of 20 candies, 5 will be red?

rnrqpr

np

red) 20 of 5(

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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nrP

)(rr pqrP 1)(

Donating BloodPage 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 can’t do it?

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NM

THE NORMAL MODEL

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

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Conditions for a Bernoulli Trial1. only 2 possibilities 2. fixed probability for the possibilities3. independent or less than 10% of the

population