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					  Probabilistic and
Statistical Techniques
           Lecture (14)

   Eng. Ismail Zakaria El Daour

              2012
                                  1
Probabilistic and Statistical Techniques




              Chapter 4 (part 1)
           Probability Distribution




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Probabilistic and Statistical Techniques


             Random Variables

    Discrete random variable


    Continuous random variable




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Probabilistic and Statistical Techniques

                  Requirements for
               Probability Distribution

                    P( x ) 1
         where x assumes all possible values

                 0  P(x)  1
            for every individual value of x

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Probabilistic and Statistical Techniques




              Chapter 4 (part 2)
           Probability Distribution




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Probabilistic and Statistical Techniques
  Mean, Variance and Standard Deviation of a
           Probability Distribution
     x.P( x )              Mean

  2   ( x   )2 .P( x )   Variance


  2  [ x2 .P( x)]  2     Variance (shortcut)


       x .P( x ) 
              2            2
                                Standard Deviation

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Probabilistic and Statistical Techniques




              Chapter 4 (part 3)
            Probability Distribution




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Probabilistic and Statistical Techniques




      Binomial Probability Distributions




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Probabilistic and Statistical Techniques


                    Key Concept
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.



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Probabilistic and Statistical Techniques
                      Definitions
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.
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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)

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Probabilistic and Statistical Techniques

                    Notation (cont)
 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.

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Probabilistic and Statistical Techniques

               Binomial distribution
        Mean, Variance & Standard deviation




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Probabilistic and Statistical Techniques

          Methods for Finding Probabilities
          Using the Binomial Probability Formula
                                            n x
                P( x) n C x  p  q  x


                x  0,1,2,...,n
 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)
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Probabilistic and Statistical Techniques

         Methods for Finding Probabilities
         Using the Binomial Probability Formula




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Probabilistic and Statistical Techniques
                           Example 1
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
                                        n x
               P( x) C  p  q
                       n x
                                  x

    P(3)5 C3  P 3  q 53
    P(3)5 C3  0.9  0.1
                              53
                                     10  0.729  0.01  0.0729
                       3



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Probabilistic and Statistical Techniques
                         Example 2
 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

                     2    1
                  1 1
                                  3
                               1 3
      P(2)  3 C2      3    .375
                  2 2      2 8


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Probabilistic and Statistical Techniques
                       Example 3
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
                                             n x
              P( x ) n C x  P  q      x


                  P(3)18 C2  P 2  q16
                  P(3)18 C2  0.1  0.9
                                     2       16



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Probabilistic and Statistical Techniques
                   Example 3 (cont.)
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

    P(3)  P(4) 18 C3  P3  q (183) 18 C4  P 4  q (184)
 Find mean & standard deviation

            np  18(0.1)  1.8
           np(1  p)  1.8(1  0.1)  1.27

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Probabilistic and Statistical Techniques
Using table to Calculate the Binomial Probability




                                   Lecture 16       21
Probabilistic and Statistical Techniques

                       Example 4
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
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Probabilistic and Statistical Techniques

 Solution:

  P(4)  P(5)
  5 C4  0.54  0.46  5 C5  0.54  0.46
              4      1            5      0




  P(0)  P(1)  P(2)


 P(2)  P(3)  P(4)  P(5)

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Probabilistic and Statistical Techniques

                         Example 5
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



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Probabilistic and Statistical Techniques

 Solution:

  P (10 )15 C10  0.810  0.25

 P(10)  P(11)  P(12)  P(13)  P(14)  P(15)
 Where success = arrive late
    OR
 P(0)  P(1)  P(2)  P(3)  P(4)  P(5)
 Where success = arrive on time

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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.



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Probabilistic and Statistical Techniques

 Solution:

  P (2) 6 C2  0.09 2 * 0.914

 P(2)  0.0833




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Probabilistic and Statistical Techniques
                       Example 7
 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

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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.



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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?



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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?



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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
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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
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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

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