Transcript downloads/larson 4 discrete

4 Discrete Probability Distributions
x = number of on time
arrivals
x = number of
correct answers
Elementary Statistics
Larson
Farber
x = number of employees
reaching sales quota
Larson/Farber Ch. 4
x = number of
points scored in a
game
Section 4.1
Probability
Distributions
Larson/Farber Ch. 4
Random Variables
A random variable, x is the numerical outcome
of a probability experiment.
x = The number of people in a car
x = The gallons of gas bought in a week
x = The time it takes to drive from home to school
x = The number of trips to school you make per week
Larson/Farber Ch. 4
Types of Random Variables
A random variable is discrete if the number of possible
outcomes is finite or countable. Discrete random variables
are determined by a count.
A random variable is continuous if it can take on any
value within an interval. The possible outcomes cannot be
listed. Continuous random variables are determined by a
measure.
Larson/Farber Ch. 4
Types of Random Variables
Identify each random variable as discrete or continuous.
x = The number of people in a car
Discrete – you count the number of people in a car 0, 1,
2, 3… Possible values can be listed.
x = The gallons of gas bought in a week
Continuous – you measure the gallons of gas. You cannot
list the possible values.
x = The time it takes to drive from home to school
Continuous – you measure the amount of time. The possible
values cannot be listed.
x = The number of trips to school you make per week
Discrete – you count the number of trips you make. The
possible numbers can be listed.
Larson/Farber Ch. 4
Discrete Probability Distributions
A discrete probability distribution lists each possible
value of the random variable, together with its probability.
A survey asks a sample
of families how many
vehicles each owns. number of
vehicles
x
0
1
2
3
P(x)
0.004
0.435
0.355
0.206
Properties of a probability distribution
• Each probability must be between 0 and 1, inclusive.
• The sum of all probabilities is 1.
Larson/Farber Ch. 4
Probability Histogram
Number of Vehicles
0.435
.40
0.355
P(x)
.30
0.206
.20
.10
0.004
0
00
11
22
33
x
• The height of each bar corresponds to the probability of x.
• When the width of the bar is 1, the area of each bar
corresponds to the probability the value of x will occur.
Larson/Farber Ch. 4
Mean, Variance and Standard Deviation
The mean of a discrete probability distribution is:
The variance of a discrete probability
distribution is:
The standard deviation of a discrete
probability distribution is:
Larson/Farber Ch. 4
Mean (Expected Value)
Calculate the mean
Multiply each value by its probability. Add the
products
x
0
1
2
3
P(x)
0.004
0.435
0.355
0.206
xP(x)
0
0.435
0.71
0.618
1.763
The expected value (the mean) is 1.763 vehicles.
Larson/Farber Ch. 4
Calculate the Variance and Standard
Deviation
The mean is 1.763 vehicles.
x
0
1
2
3
P(x)
0.004
0.435
0.355
0.206
x- μ
-1.763
-0.763
0.237
1.237
(x - μ )
3.108
0.582
0.056
1.530
P(x)(x P(x)
- )
0.012
0.253
0.020
0.315
0.601
variance
The standard deviation is 0.775 vehicles.
Larson/Farber Ch. 4
Section 4.2
Binomial Distributions
Larson/Farber Ch. 4
Binomial Experiments
Characteristics of a Binomial Experiment
• There are a fixed number of trials. (n)
• The n trials are independent and repeated under identical
conditions.
• Each trial has 2 outcomes,
S = Success or F = Failure.
• The probability of success on a single trial is p. P(S) = p
The probability of failure is q. P(F) =q where p + q = 1
• The central problem is to find the probability of x
successes out of n trials. Where x = 0 or 1 or 2 … n.
The random variable x is a count of the
number of successes in n trials.
Larson/Farber Ch. 4
Guess the Answers
1. What is the 11th digit after the decimal point for the irrational number e?
(a) 2
(b) 7
(c) 4
(d) 5
2. What was the Dow Jones Average on February 27, 1993?
(a) 3265
(b) 3174
(c) 3285
(d) 3327
3. How many students from Sri Lanka studied at U.S. universities
from 1990-91?
(a) 2320
(b) 2350
(c) 2360
(d) 2240
4. How many kidney transplants were performed in 1991?
(a) 2946
(b) 8972
(c) 9943
(d) 7341
5. How many words are in the American Heritage Dictionary?
(a) 60,000
(b) 80,000
(c) 75,000
(d) 83,000
Larson/Farber Ch. 4
Quiz Results
The correct answers to the quiz are:
1. d
2. a
3. b
4. c
5. b
Count the number of correct answers. Let the
number of correct answers = x.
Why is this a binomial experiment?
What are the values of n, p and q?
What are the possible values for x?
Larson/Farber Ch. 4
Binomial Experiments
A multiple choice test has 8 questions each of which has 3
choices, one of which is correct. You want to know the
probability that you guess exactly 5 questions correctly.
Find n, p, q, and x.
n=8
p = 1/3
q = 2/3
x=5
A doctor tells you that 80% of the time a certain type of surgery is
successful. If this surgery is performed 7 times, find the probability
exactly 6 surgeries will be successful. Find n, p, q, and x.
n=7
p = 0.80
Larson/Farber Ch. 4
q = 0.20
x=6
Binomial Probabilities
Find the probability of getting exactly 3 questions correct on the quiz.
Write the first 3 correct and the last 2 wrong as SSSFF
P(SSSFF) = (.25)(.25)(.25)(.75)(.75) = (.25)3(.75)2 = 0.00879
Since order does not matter, you could get any combination
of three correct out of five questions. List these
combinations.
SSSFF
FFSSS
SSFSF SSFFS SFFSS SFSFS
FSFSS FSSFS SFSSF FFSSF
Each of these 10 ways has a probability of 0.00879.
P(x = 3) = 10(0.25)3(0.75)2 = 10(0.00879) = 0.0879
Larson/Farber Ch. 4
Combination of n values, choosing x
There are
ways.
Find the probability of getting exactly 3 questions correct on the
quiz.
Each of these 10 ways has a probability of 0.00879.
P(x = 3) = 10(0.25)3(0.75)2= 10(0.00879)= 0.0879
Larson/Farber Ch. 4
Binomial Probabilities
In a binomial experiment, the probability of exactly x
successes in n trials is
Use the formula to calculate the probability of getting none correct,
exactly one, two, three, four correct or all 5 correct on the quiz.
P(3) = 0.088
Larson/Farber Ch. 4
P(4) = 0.015
P(5) = 0.001
Binomial Distribution
x
0
1
2
3
4
5
Binomial Histogram
.396
.40
.30
.294
.237
P(x)
0.237
0.396
0.264
0.088
0.015
0.001
.20
.088
.10
.015
.001
4
5
0
0
Larson/Farber Ch. 4
1
2
3
x
Probabilities
1. What is the probability of answering
either 2 or 4 questions correctly?
P( x = 2 or x = 4) = 0.264 + 0.015 = 0. 279
x
0
1
2
3
4
5
P(x)
0.237
0.396
0.264
0.088
0.015
0.001
2. What is the probability of answering at least 3 questions correctly?
P(x  3) = P( x = 3 or x = 4 or x = 5) = 0.088 + 0.015 + 0.001 = 0.104
3. What is the probability of answering at least one question correctly?
P(x  1) = 1 - P(x = 0) = 1 - 0.237 = 0.763
Larson/Farber Ch. 4
Parameters for a Binomial Experiment
Mean:
Variance:
Standard deviation:
Use the binomial formulas to find the mean, variance and
standard deviation for the distribution of correct answers on
the quiz.
Larson/Farber Ch. 4
Section 4.3
More Discrete
Probability Distributions
Larson/Farber Ch. 4
The Geometric Distribution
A marketing study has found that the probability that a person who
enters a particular store will make a purchase is 0.30.
•The probability the first purchase will be made by the first person
who enters the store 0.30. That is P(1) = 0.30.
•The probability the first purchase will be made by the second
person who enters the store is (0.70) ( 0.30). So P(2) = (0.70) ( 0.30)
= 0.21.
•The probability the first purchase will be made by the third person
who enters the store is (0.70)(0.70)( 0.30). So P(3) = (0.70) (0.70)
(0.30) = 0.147.
The probability the first purchase will be made by
person number x is P(x) = (.70)x - 4(.30)
Larson/Farber Ch. 4
The Geometric Distribution
A geometric distribution is a discrete probability distribution
of the random variable x that satisfies the following
conditions.
1. A trial is repeated until a success occurs.
2. The repeated trials are independent of each
other.
3. The probability of success p is the same for
each trial.
The probability that the first success will occur on trial
number x is
P(x) = (q)x – 1p
where q = 1 – p
Larson/Farber Ch. 4
Application
A cereal maker places a game piece in its boxes. The
probability of winning a prize is one in four. Find the
probability you
a) Win your first prize on the 4th purchase
P(4) = (.75)3 .(.25) = 0.1055
b) Win your first prize on your 2nd or 3rd purchase
P(2) = (.75)1(.25) = 0.1875 and
P(3) = (.75)2(.25) = 0.1406
So P(2 or 3 ) = 0.1875 + 0.1406 = 0.3281
c) Do not win your first prize in your first 4 purchases.
1 – (P(1) + P(2) + P(3) + P(4))
1 – ( 0.25 + 0.1875 + 0.1406 + 0.1055)
= 1 – .6836 = 0.3164
Larson/Farber Ch. 4
The Poisson Distribution
The Poisson distribution is a discrete probability distribution
of the random variable x that satisfies the following
conditions.
1. The experiment consists of counting the number of times, x, an
event occurs in an interval of time, area or space.
2. The probability an event will occur is the same for each interval.
3. The number of occurrences in one interval is independent of the
number of occurrences in other intervals.
The probability of exactly x occurrences in an interval is
e is the irrational number approximately 2.71828
is the mean number of occurrences per interval.
Larson/Farber Ch. 4
Application
It is estimated that sharks kill 10 people each
year worldwide. Find the probability
a) Three people are killed by sharks this year
b) Two or three people are killed by sharks this
year
P(3) = 0.0076
P(2 or 3) = 0.0023 + 0.0076 = 0.0099
Larson/Farber Ch. 4