Geometric Geometric DistributionDistribution

Similar to BinomialSimilar to Binomial• Success/FailureSuccess/Failure• Probabilities do NOT changeProbabilities do NOT change

Now you are looking at the number of Now you are looking at the number of failures until a success.failures until a success.

Determining the prob2Determining the prob2

Probability and Expectation for Probability and Expectation for Geometric DistributionGeometric Distribution

Where p is the Where p is the probability of a probability of a success in each success in each single trial and q is single trial and q is the probability of a the probability of a failurefailure

The expectation The expectation converges to a converges to a simple formulasimple formula

( ) xP x q p ( )q

E xp

ExEx

Jamaal has a success rate of 68% for Jamaal has a success rate of 68% for scoring on free throws in basketball. scoring on free throws in basketball. What is the expected waiting time What is the expected waiting time before he misses the basket on a before he misses the basket on a free throw?free throw?

The random variable is the number of The random variable is the number of trials before he misses a free throwtrials before he misses a free throw

A success is Jamaal failing to score A success is Jamaal failing to score

q=0.68q=0.68

p=1-0.68=0.32p=1-0.68=0.32

( )

0.68

0.322.1

qE X

p

ExEx

Suppose that an intersection you pass on Suppose that an intersection you pass on your way to school has a traffic light your way to school has a traffic light that is green 40 s and then amber or that is green 40 s and then amber or red for a total of 60sred for a total of 60s

a)a) What is the probability that the light What is the probability that the light will be green when you reach the will be green when you reach the intersection at least once a week?intersection at least once a week?

b)b) What is the expected number of days What is the expected number of days before the light is green when you before the light is green when you reach the intersection?reach the intersection?

a)a) What is the probability that the light will be green What is the probability that the light will be green when you reach the intersection at least once when you reach the intersection at least once a week?a week?

p= light is green = 40/100 = 0.40p= light is green = 40/100 = 0.40

q= light not green = 60/100 = 0.60q= light not green = 60/100 = 0.60

There are 5 school days so we want There are 5 school days so we want the probability that you will wait 0 the probability that you will wait 0 days, 1 day , 2 days, 3 days or 4 days, 1 day , 2 days, 3 days or 4 days before it is greendays before it is green

0.92

2 3

4

(0,1, 2,3 4)

0.40 (0.6)(0.4) (0.6) (0.4) (0.6) (0.4)

(0.6) (0.4)

P or

b) What is the expected number of days before the light is b) What is the expected number of days before the light is green when you reach the intersection?green when you reach the intersection?

The expected The expected waiting time waiting time before catching a before catching a green light is 1.5 green light is 1.5 daysdays

( )

0.6

0.41.5

qE X

p

Homework!Homework!

Pg 394Pg 394

#1,2,3,7,9,10 #1,2,3,7,9,10