• 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 6.3

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.• HOW IS THIS DIFFERENT FROM A BINOMIAL???In a binomial setting, the number of 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 how many successes you get in n trials.

So, the random variable is how many trials it takes to 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

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!

ppnXP n 1)1()(

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

npnxP )1()(

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

• The formula for the variance of a

geometric RV is

1X p

22

(1 )X

p

p

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