Transcript prob_stats_ch04

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.
1. x = The number of people in a car
2. x = The gallons of gas bought in a week
3. x = The time it takes to drive from home to school
4. x = The number of trips to school you make per week
Larson/Farber Ch. 4
Discrete Probability Distributions
A discrete probability distribution
lists each possible value of the
# of vehicles
random variable with
x
its probability.
A survey asks a sample
of families how many
vehicles each owns.
0
1
2
3
Probability
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:
The variance of a discrete probability distribution:
The standard deviation of a discrete probability
distribution:
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)
The expected value (the mean) is ____________ vehicles.
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
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
(x - μ )
P(x)(x P(x)
- )
variance
The standard deviation is __________ vehicles.
Larson/Farber Ch. 4
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)(xP(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 independent trials (n)
• Each trial has 2 outcomes,
S = Success or F = Failure.
• The probability of success is the same for each trial
• The random variable x counts the number of successful
trials
Larson/Farber Ch. 4
Binomial Experiments
Notation for Binomial Experiments
• n
the number of times the trial is repeated
• x
the random variable represents a count of the number
of successes in n trials: x = 0,1,2,3,…..n
• p = P(S)
the probability of success in a single trial
• q = P(F)
the probability of failure in a single trial
p+q=1
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.
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.
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
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 Probabilities
Find the probability of getting exactly 3 correct on the quiz you took earlier.
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
FSFSS
SSFFS
FSSFS
SFFSS
SFSSF
SFSFS
FSSSF
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
= 10 ways
Find the probability of getting exactly 3 correct on the quiz.
P(x = 3) = 10(0.25)3(0.75)2= 10(0.00879)= 0.0879
In a binomial experiment, the probability of exactly x
successes in n trials is
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.
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?
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 getting at least 3 questions correct?
3. What is the probability of getting at least one question correct?
Larson/Farber Ch. 4
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 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
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 - 1(.30)
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
b) Win your first prize on your 2nd or 3rd purchase
c) Do not win your first prize in your first 4 purchases.
Larson/Farber Ch. 4
Application
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
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.
(.75)4 = 0.3164 or
1 – (P(1) + P(2) + P(3) + P(4))
1 – ( 0.25 + 0.1875 + 0.1406 + 0.1055) = 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. Counting the number of times, x, an event occurs in an
interval of time, area or volume.
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 approximately 2.718
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