Download Geometric Distributions

AP STATS: Warm-Up
Flip the quarter on your desk 10 times. Let X be the
the number of heads. Is this a binomial distribution?
What is the probability of observing exactly 4 heads?
At MOST 4 heads?
Here’s the Scenario
• Roll a die until you see a 3. How many rolls
does it take until you observe your first 3.
Repeat this three times.
8.2 The Geometric Distribution
1.
2.
3.
4.
What is the geometric setting?
How do you calculate the probability of getting
the first success on the nth trial?
How do you calculate the means and variance of
a geometric distribution?
How do you calculate the probability that it takes
more than n trials to see the first success for a
geometric random variable?
The Geometric Distribution
A geometric random variable is defined as
X = number of trials until the first success is observed (including
the success trial)
The probability distribution of x is called the geometric
probability distribution.
Example: Rolling a Die
• A game consists of rolling a die. The event of
interest is rolling a 3. The event is called a
success. The random variable X = the number
of trials until a 3 occurs.
Calculating Geometric Probabilities
Example: Rolling a Die
A game consists of rolling a die. The event of interest is
rolling a 3. The event is called a success. The random
variable X = the number of trials until a 3 occurs.
What is the probability that the first 3 will occur first roll?
2nd? 3rd? 4th? 5th? 6th? 7th?
Construct a probability distribution
You Try!
• Over a very long period of time, it has
been noted that on Friday’s 25% of the
customers at the drive-in window at the
bank make deposits.
• What is the probability that it takes 4
customers at the drive-in window before
the first one makes a deposit.
Example - solution
• This problem is a geometric distribution
problem with  = 0.25.
• Let x = number of customers at the drivein window before a customer makes a
deposit.
The desired probability is
p(4)  (.75)41(.25)  0.0117
Geometric Probability
Probabilit y of success on nth trial 
P ( n)  p (1  p ) n 1
mean   
1
p
Variance   2 
standard deviation   
1 
2
1- p
p
The Mean of a Geometric Distribution is the Expected Value
(i.e. how long we can expect it to take before our first success.)