Transcript Requirements of geometric distributions

CHS Statistics
Objective: To solve multistep probability
tasks with the concept of geometric
distributions
Geometric Distributions

 A Geometric probability model tells us the
probability for a random variable that counts the
number of trials until the first success.
Geometric Distributions

Bernoulli Trials
Requirements of geometric distributions:
1.
Each observation is in one of two categories: success or
failure.
2.
The probability is the same for each observation.
3.
Observations are independent. (Knowing the result of one
observation tells you nothing about the other observations.)
4.
The variable of interest is the number of trials required to
obtain the first success. [The only difference from a binomial
distribution]
Example

Does this represent a geometric distribution? What is
your evidence?
 A new sales gimmick is to sell bags of candy that have
30% of M&M’s covered with speckles. These “groovy”
candies are mixed randomly with the normal candies as
they are put into the bags for distribution and sale. You
buy a bag and remove candies one at a time looking for
the speckles.
Geometric Model

A new sales gimmick is to sell bags of candy that have 30% of M&M’s
covered with speckles. These “groovy” candies are mixed randomly with
the normal candies as they are put into the bags for distribution and sale.
You buy a bag and remove candies one at a time looking for the speckles.
 What’s the probability that the first speckled one we see is the fourth candy
we get? Note that the skills to answer this question come from the very first
day of the probability unit.
Geometric Model (cont.)

 What’s the probability that the first speckled one is the tenth one?
Write a general formula.
 What’s the probability that the first speckled candy is one of the first
three we look at?
 How many do we expect to have to check, on average, to find a
speckled one?
Geometric Model (cont.)

p = probability of success
q = 1 – p = probability of failure
X = number of trials until the first success occurs
x-1
P(X = x) = q p
1
E(X)   
p

q
p2
Example

 People with O-negative blood are “universal donors.” Only about 6% of
people have O-negative blood.
1.
If donors line up at random for a blood drive, how many do you
expect to examine before you find someone who has O-negative
blood?
2.
What’s the probability that the first O-negative donor found is one of
the four people in line?
Geometric Probabilities Using
Calculator

 2nd  DISTR  geometpdf(
 Note the pdf for Probability Density Function
 Used to find any individual outcome
 Format: geometpdf(p,x)
 2nd  DISTR  geometcdf(
 Note the cdf for Cumulative Density Function
 Used to find the first success on or before the xth trial
 Format: geometcdf(p,x)
 Try the last example using the calculator! Much easier…
Example

Example: Let x represent the number of students who must be stopped
before finding one with jumper cables. Suppose 40% of students who
drive to school carry cables. Find the probability that the
 3rd person you stop has them.
 You need to stop no more than 3 people.
Assignment

 Pp. 401-404 # 10 – 22 Even
 Be sure to check your answers with the solutions
manual online.