1. Probabilistic and Statistical Techniques
Lecture 17 Eng. Ismail Zakaria El Daour
2010

2. Chapter 4 (part 1) Probability Distribution
Probabilistic and Statistical Techniques
Chapter 4 (part 1) Probability Distribution 2

3. Random Variables Discrete random variable Continuous random variable
Probabilistic and Statistical Techniques
Random Variables
Discrete random variable
Continuous random variable

4. Requirements for Probability Distribution
Probabilistic and Statistical Techniques
Requirements for Probability Distribution
where x assumes all possible values
Page 203 of Elementary Statistics, 10th Edition
0  P(x)  1
for every individual value of x

5. Chapter 4 (part 2) Probability Distribution
Probabilistic and Statistical Techniques
Chapter 4 (part 2) Probability Distribution 5

6. Mean, Variance and Standard Deviation of a Probability Distribution
Probabilistic and Statistical Techniques
Mean, Variance and Standard Deviation of a
Probability Distribution
Mean
Variance
Variance (shortcut)
Standard Deviation
In Chapter 3, we found the mean, standard deviation ,variance, and shape of the distribution for actual observed experiments.
The probability distribution and histogram can provide the same type information.
These formulas will apply to ANY type of probability distribution as long as you have all the P (x) values for the random variables in the distribution.
In section 4 of this chapter, there will be special EASIER formulas for the special binomial distribution.
The TI-83 and TI-83 Plus calculators can find the mean, standard deviation, and variance in the same way that one finds those values for a frequency table.
With the TI-82, TI-81, and TI-85 calculators, one would have to multiply all decimal values in the P(x) column by the same factor so that there were no decimals and proceed as usual.
Page 204 of Elementary Statistics, 10th Edition

7. Chapter 4 (part 3) Probability Distribution
Probabilistic and Statistical Techniques
Chapter 4 (part 3) Probability Distribution 7

8. Binomial Probability Distributions
Probabilistic and Statistical Techniques
Binomial Probability Distributions 8

9. Key Concept Probabilistic and Statistical Techniques
This section presents a basic definition of a binomial distribution along with notation, and it presents methods for finding probability values.
Binomial probability distributions allow us to deal with circumstances in which the outcomes belong to two relevant categories such as acceptable/defective or survived/died. 9

10. Definitions Probabilistic and Statistical Techniques
A binomial probability distribution results from a procedure that meets all the following requirements:
1. The procedure has a fixed number of trials.
2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.)
3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure).
4. The probability of a success remains the same in all trials. 10

12. Notation for Binomial Probability Distributions
Probabilistic and Statistical Techniques
Notation for Binomial Probability Distributions
S and F (success and failure) denote two possible categories of all outcomes; p and q will denote the probabilities of S and F, respectively, so
P(S) = p (p = probability of success)
P(F) = 1 – p = q (q = probability of failure) 12

13. Notation (cont) Probabilistic and Statistical Techniques
n denotes the number of fixed trials.
x denotes a specific number of successes in n trials, so
x can be any whole number between 0 and n, inclusive.
p denotes the probability of success in one of the n trials.
q denotes the probability of failure in one of the n trials.
P(x) denotes the probability of getting exactly x successes
among the n trials. 13

14. Binomial distribution Mean, Variance & Standard deviation
Probabilistic and Statistical Techniques
Binomial distribution Mean, Variance & Standard deviation

15. Methods for Finding Probabilities
Probabilistic and Statistical Techniques
Methods for Finding Probabilities
Using the Binomial Probability Formula
where
n = number of trials
x = number of successes among n trials
p = probability of success in any one trial
q = probability of failure in any one trial (q = 1 – p)
page 216 of Elementary Statistics, 10th Edition

16. Methods for Finding Probabilities
Probabilistic and Statistical Techniques
Methods for Finding Probabilities
Using the Binomial Probability Formula

17. Example 1 Probabilistic and Statistical Techniques
Use the binomial probability formula to find the probability of getting exactly 3 correct responses among 5 different requests from directory assistance.
Assume that in general the responses is correct 90% of the time.
That is Find P(3) given that n=5, x=3, p=0.9 & q=0.1

18. Example 2 Probabilistic and Statistical Techniques
Consider the experiment of flipping a coin 3 times. If we let the event of getting tails on a flip be considered “success”, and if the random variable T represents the number of tails obtained, then T will be binomially distributed with n=3 ,p=0.5 , and q=0.5 .
calculate the probability of exactly 2 tails

19. Example 3 Probabilistic and Statistical Techniques
Each sample of water has a 10% chance of containing a particular organic pollutant. Assume that the samples are independent with regard to the presence of the pollutant.
Find the probability that in the next 18 samples, exactly 2 contain the pollutant.
Let n=18, x=2, p=0.1 & q=0.9

20. Example 3 (cont.) Probabilistic and Statistical Techniques Also,
Find the probability that in the next 18 samples, that 3 or 4 contain the pollutant.
Let n=18, p=0.1 & q=0.9
Find mean & standard deviation

21. Using table to Calculate the Binomial Probability
Probabilistic and Statistical Techniques
Using table to Calculate the Binomial Probability
Lecture 16

22. Example 4 Probabilistic and Statistical Techniques
There is a 0.54 probability that a randomly selected freshman at a two-year college will return the second year. In each case, assume that 5 freshmen at a two-year college are randomly selected and find the probability indicated
Find the probability that at least four of the freshmen return for the second year
Find the probability that at most two of the freshmen return for the second year
Find the probability that more than one of the freshmen return for the second year 22

23. Probabilistic and Statistical Techniques
Solution: 23

24. Example 5 Probabilistic and Statistical Techniques
The rate of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best rate with 80% of its flights arriving on time. A test is conducted by randomly selecting 15 southwest flights and observing whether they arrive on time.
Find the probability that exactly 10 flights arrive on time
Find the probability that at least 10 flights arrive late 24

25. Probabilistic and Statistical Techniques
Solution:
Where success = arrive late OR
Where success = arrive on time 25

26. Probabilistic and Statistical Techniques
Example 6
Nine Percent of men and 0.25% of women cannot distinguish between the colors red and green. This is a type of color blindness that cause problems with traffic signals. If 6 men are randomly selected for a study of traffic signals perceptions, find the probability that exactly two of them cannot distinguish between red and green. 26

27. Probabilistic and Statistical Techniques
Solution: 27

28. Example 7 Probabilistic and Statistical Techniques
In a housing study, it was found that 26% of college students live in campus housing. The providence Insurance Company wants to sell those students special policies insuring their personal property. If they test a marketing strategy by randomly selecting six college students,
what is the probability that at least one of them live in campus housing
what is the mean of no. of students live in campus housing 28

29. Probabilistic and Statistical Techniques
Example 8
Air America has a policy of booking as many as 15 persons on an airplane that can seat only 14. Because past studies have revealed that only 85% of the booked passengers actually arrive for the flight. Find the Probability that if Air America books 15 persons, not enough seats will be available. 29

30. Probabilistic and Statistical Techniques
Example 9
An automobile manufacturer has determined that 30% of all gas tanks that were installed on its 2002 compact model are defective. If 14 of these cars are independently sampled, what is the probability that more than 11 cars need new gas tanks? 30

31. Probabilistic and Statistical Techniques
Example 10
There are five flights daily from Pittsburgh via US airways into the Bradford, Pennsylvania Regional Airport. Suppose the probability that any flight arrives late is 0.2. what is the probability that none of the flights are late today? What is the probability that exactly one of the flights is late today? 31

32. Probabilistic and Statistical Techniques
Example 11
Suppose 60 percent of people prefer Coke to Pepsi. We select 18 people for further study.
How many would you expect to prefer Coke ?
What is the probability 10 of those surveyed will prefer Coke ?
What is the probability 15 prefer Coke ?
Compute the mean and the standard deviation 32

33. Probabilistic and Statistical Techniques
Example 12
In a recent study 90% of the homes in the United States were found to have large screen TVs. In a sample of nine homes, what is the probability that:
All nine have large screen TVs
Less than five have large screen TVs
More than five have large screen TVs
at least seven homes have large screen TVs 33

34. Probabilistic and Statistical Techniques
Example 13
A company receive 60% of its order over the internet. Within a collection of 18 independently placed orders. What is the probability that:
Between eight and ten (inclusive) of the orders are received over the internet
no more than four of the orders are received over the internet 34

35. Thanks for Your Attention