Transcript Week_3_Lecture_1_ILS

MATH 221
Integrated Learning System
Week 3, Lecture 1
More Discrete Probability
Distributions
Objectives:
At the end of this presentation you should be able to find
probabilities using the geometric distribution and the
Poisson distribution.
Definition
A geometric distribution is a discrete probability
distribution of a random variable x that satisfies the
following conditions:
1. The trial is repeated until a success occurs.
2. The repeated trials are independent.
3. The probability of success p is constant for each
trial.
Geometric Probability Function
P(x) = pqx-1 where q = 1 – p.
Geometric Probability Example
A cereal maker places a game piece in its cereal boxes.
The probability of winning a prize in the game is ¼.
Find the probability that you:
1. Win your first prize with your 4th purchase.
2. Win your first prize with your 1st, 2nd, or 3rd purchase.
3. You do not win a prize with your first four purchases.
(Problem 10, page 193, Larson and Farber)
Geometric Distribution
So, what does the distribution we have just used look
like?
Geometric Distribution for p = 0.25
0.3
0.25
P (x )
0.2
0.15
0.1
0.05
0
1
2
3
4
5
6
x
7
8
9
10
Using Technology: TI-83
geometpdf(p,x) – Probability density function
geometcdf(p,x) – Cumulative probability density function
Using Technology: EXCEL
While EXCEL contains many of the probability density
functions that we will use in this course, it does not have
a geometric probability density function. However, the
geometric probability density function is fairly simple and
it is easy to enter as a cell function.
Exercise
Basketball Player Shaquille O’Neal makes a free throw
shot about 53.4% of the time. Find the probability
that:
1. The first shot O’Neal makes is the second shot.
2. The first shot O’Neal makes is the first or second
shot.
3. O’Neal does not make two shots.
Definition
The Poisson distribution is a discrete probability
distribution that satisfies the following
1. The experiment consists of counting the number of
times x that an event occurs in a given interval.
2. The probability of the event is the same for each
interval.
3. The number of occurrences is independent from
interval to interval.
Poisson Probability Function
P x  
x
 e

x!
, in some texts written P x

x
 e

x!
In the versions of the function above  and  represent
the mean arrival rate or rate of occurrence for the event
of interest.
 is the lower case Greek letter mu,  is the lower case
Greek letter lambda, and e is the base of the natural
logarithms.
Example
A newspaper finds that the mean number of
typographical errors per page is four. Find the
probability that:
1. Exactly 3 typographical errors will be found on a
page.
2. At most 3 typographical errors will be found on a
page.
3. More than 3 typographical errors will be found on a
page.
Example (Solution)
4
3
4 e
1.
P 3  
2.
P x  3   P 0   P 1   P 2   P 3 
 0 . 195
3!
0

4 e
4
0!

4e
4
1!
2

4 e
2!
4
3

4 e
4
3!
 0 . 018  0 . 073  0 . 147  0 . 195
 0 . 433
3.
P  x  3   1  P  x  3   1  0 . 433  0 . 567
Using Technology: TI-83
poissonpdf(,x) – Probability density function
poissoncdf(,x) – Cumulative probability density
function
Using Technology: EXCEL
Using Technology: EXCEL
Using Technology: EXCEL
Poisson Distribution for mu of 4
0.25
P (x )
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
x
6
7
8
9
10
Exercise
A major hurricane is a hurricane with wind speeds of 111
mph or greater. From 1900 to 1999, the mean
number of major hurricanes to strike the U.S.
mainland per year was about 0.6. Find the
probability that in a given year:
1. Exactly one major hurricane will strike the U.S.
2. At most one major hurricane will strike the U.S.
3. More than one major hurricane will strike the U.S.