Transcript Section 8.2

• Your mail-order company advertises
that it ships 90% of its orders within
three working days. You select an
SRS of 100 of the 5000 orders
received in the past week for an
audit. The audit reveals that 86 of
these orders were shipped on time.
• If the company really ships 90% of its
orders on time, what is the probability
that 86 or fewer in an SRS of 100 orders
are shipped on time?
• Can you claim that the company is not
telling the truth?
Section 8.2
Geometric Random
Variables
Discrete Random
Variables
• Recall that discrete random
variables have a countable number
of possible values.
• One special class of discrete random
variables is the binomial distribution.
Do you remember the four
characteristics of a binomial
setting?
Geometric Random
Variables
• The geometric random variable is another
discrete random variable.
• There are also four conditions in a
geometric setting.
• Each observation has two outcomes (success or
failure).
• The probability of success is the same for each
observation.
• The observations are all independent.
• The variable of interest is how many trials are
required to obtain the first success.
•InHOW
IS THISsetting,
DIFFERENT
FROM A
a binomial
the number
ofBINOMIAL???
trials is
fixed; the variable of interest is how many
successes there are.
Comparison of Binomial
to Geometric
Binomial
Geometric
Each observation has two
outcomes (success or
failure).
Each observation has two
outcomes (success or
failure).
The probability of success is
the same for each
observation.
The probability of success is
the same for each
observation.
The observations are all
independent.
The observations are all
independent.
There are a fixed number of
trials.
There is a fixed number of
successes (1).
So, the random variable is
So, the random variable is
how many successes you get how many trials it takes to
in n trials.
get one success.
Is this geometric?
• In the game of “Trouble”, you
need to roll a 6 on a standard
die to get started. What is the
probability that it takes more
than 6 rolls to get a six?
Is this geometric?
• I’m going to roll a die 10 times
and see how many times I get a
6. What is the probability that I
get at least 5 6s?
How to Calculate
Geometric Probabilities
• It’s usually not difficult to
calculate these by hand.
• Let’s take the “Trouble” game
example.
• What is the P(X = 4)? That means
what is the probability that the first
six occurs on the 4th roll?
• How can we use this idea to get a
general formula?
Calculating Geometric
Probabilities
P( X  n)  (1  p)
n1
p
This is the probability that you
have successes on the nth trial.
p is the probability of success.
This is NOT on your formula
sheet!
Example
• An experiment consists of rolling a
die until a prime number (2, 3, 5) is
observed.
• Define the random variable X.
• Verify that it is geometric.
• What is the probability that you roll your
first prime on the first roll.
• What is the probability that you roll your
first prime on the 8th roll?
Let’s Construct a Probability
Distribution for a Geometric
Random Variable
• Suppose Shaq is a 40% free
throw shooter. Begin
constructing a probability
distribution for how many shots
it takes for him to make his first
free throw (let’s go to n = 10).
• What would the graph of a
geometric probability distribution
look like?
Putting it all together…
• We’ve studied two large
categories of random variables:
discrete and continuous.
• Among the discrete RVs, we’ve
studied the binomial and
geometric RVs.
• The graph of a binomial RV can be
skewed left, symmetric, or skewed
right, depending on the value of p.
• The graph of a geometric RV is
ALWAYS skewed right. Always.
• Other discrete RVs can be given to
you in the form of a table.
Continued
• Among the continuous, we’ve
studied the uniform and normal
RVs.
• To find probabilities of a uniform RV,
use geometry. Other continuous RVs
can be given to you in the form of a
graph.
• To find probabilities of a normal RV,
convert to a Z score and use Table A.
P(X > n)
• The probability that it takes
more than n trials to see the
first success is
P( x  n)  (1  p)
n
Example
• The State Department is trying to
identify an individual who speaks
Farsi (similar to Arabic) to fill a
foreign embassy position. They have
determined that 4% of the applicant
pool are fluent in Farsi.
• What is the probability that they will
have to interview more than 25 until
they find one who speaks Farsi? More
than 40?
Mean and Variance of
Geometric RV
• The formula for the mean of a
geometric RV is
1
X 
p
• The formula for the variance of a
geometric RV is
(1  p)
 
2
p
2
X
Example
• Let’s play “Trouble” again.
Remember that I need to roll a 6
in order to begin the game. The
random variable X is the number
of rolls it takes to get my first 6.
• What is the expected value of
X?
• What is the standard deviation
of X?
Simulating Geometric RVs
• Geometric distributions are often
called “waiting time” simulations;
i.e. how long must you “wait” until
you roll your first 6?
• Conducting a geometric simulation
by hand is tedious but easy.
More Trouble
• Let’s simulate rolling a six for the
game of “Trouble.”
• Which digits will represent a success?
• Which digits will represent a failure?
• Will we ignore any digits?
• How will we define the end of a trial?
• What variable will we measure?
• Your group should conduct 20 trials
of this simulation. Record your
results on the board in a probability
distribution.
Recall…
• What are the four conditions of
a geometric setting?
• What are the four conditions of
a binomial setting?
• What is the shape of a
geometric random variable’s
graph?
• What is the shape of a binomial
random variable’s graph?
Homework
Chapter 6
# 95, 96, 97, 98, 99