Chapter 9 Counting Principles Further Probability Topics

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Chapter 9Counting Principles Further
Probability Topics

• 9.1 Probability Distributions Expected
  Value
• 9.2 Multiplication Principle,
  Permutations, Combinations
• 9.4 Binomial Probability

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9.1 Probability Distributions Expected

.......Value
Random Variables

• A random variable is a numerical description of
  the outcome of an experiment.
• A random variable can be classified as being
  either discrete or continuous depending on the
  numerical values it assumes.
• A discrete random variable may assume either a
  finite number of values or an infinite sequence
  of values.
• A continuous random variable may assume any
  numerical value in an interval or collection of
  intervals.

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Discrete Random Variables

• Discrete random variable with a finite number of
  values
• Let x number of TV sets sold at the store in
  one day
• where x can take on 5 values (0, 1, 2, 3,
  4)
• Discrete random variable with an infinite
  sequence of values
• Let x number of customers arriving in one day
• where x can take on the values 0, 1, 2, .
  . .
• We can count the customers arriving, but there
  is no finite upper limit on the number that might
  arrive.

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Discrete Probability Distributions

• The probability distribution for a random
  variable describes how probabilities are
  distributed over the values of the random
  variable.
• The probability distribution is defined by a
  probability function, denoted by f(x), which
  provides the probability for each value of the
  random variable.
• The required conditions for a discrete
  probability function are
• f(x) 0
• ?f(x) 1
• We can describe a discrete probability
  distribution with a table, graph, or equation.

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Example JSL Appliances

• Using past data on TV sales (below left), a
  tabular representation of the probability
  distribution for TV sales (below right) was
  developed.
• Number
• Units Sold of Days x P(x)
• 0 80 0 .40
• 1 50 1 .25
• 2 40 2 .20
• 3 10 3 .05
• 4 20 4 .10
• 200 1.00

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Example JSL Appliances

• Graphical Representation of the Probability
  Distribution (Histogram)

.50
.40
.30
Probability
.20
.10
0 1 2 3 4
Values of Random Variable x (TV sales)
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Example JSL Appliances

• Graphical Representation of the Probability
  Distribution (Histogram)

.50
.40
.30
Probability
.20
.10
0 1 2 3 4
Values of Random Variable x (TV sales)
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Discrete Uniform Probability Distribution

• The discrete uniform probability distribution is
  the simplest example of a discrete probability
  distribution given by a formula.
• The discrete uniform probability function is
• P(x) 1/n
• where
• n the number of values the random
• variable may assume
• Note that the values of the random variable are
  equally likely.

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Expected Value

• The expected value, or average, of a random
  variable is a measure of its central location.
• Expected value of a discrete random variable
• E(x) ?xiP(xi)

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Example JSL Appliances

• Expected Value of a Discrete Random Variable
• x P(x) xP(x)
• 0 .40 .00
• 1 .25 .25
• 2 .20 .40
• 3 .05 .15
• 4 .10 .40
• 1.20 E(x)
• The expected number of TV sets sold in a day is
  1.2

?xP(x)
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9.2 The Multiplication Principle, Permutations,
Combinations

• Multiplication Principle For experiments
  involving multiple steps.
• Combinations For experiments in which r elements
  are to be selected from a larger set of n
  objects.
• Permutations For experiments in which r elements
  are to be selected from a larger set of n
  objects, where the order of selection is
  important.

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A Counting Rule for Multiple-Step Experiments

• THE MULTIPLICATION PRINCIPLE
• If an experiment consists of a sequence of k
  steps in which there are n1 possible results for
  the first step, n2 possible results for the
  second step, and so on, then the total number of
  experimental outcomes is given by (n1)(n2)...(nk).

Example Roll a 6-sided die, and toss a coin. How
many experimental outcomes are possible?
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Example Multiplication Principle

• 2-Step Experiment

(1,H)
H
T
(1,T)
n1 6
(2,H)
1
H
T
(2,T)
2
(3,H)
H
T
3
(3,T)
(4,H)
4
H
T
(4,T)
5
H
(5,H)
T
6
(5,T)
H
(6,H)
T
(6,T)
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Example Multiplication Principle

• 2-Step Experiment

(1,H)
H
T
(1,T)
n2 2
(2,H)
1
H
T
(2,T)
2
(3,H)
H
T
3
(3,T)
6 ? 2 12
(4,H)
4
H
T
(4,T)
5
H
(5,H)
T
6
(5,T)
H
(6,H)
T
(6,T)
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Counting Rule for Combinations

• Another useful counting rule enables us to count
  the number of experimental outcomes when r
  elements are to be selected from a larger set of
  n objects.
• The number of combinations of n elements taken r
  at a time is

• FACTORIAL NOTATION
• For any natural number n
• n! n(n - 1)(n - 2)
  . . . (3)(2)(1)
• Also,
• 0! 1

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Combination - Example

• In the Florida Lottery contestants choose 6
  numbers from 1 to 53. How many possible
  combinations of 6 numbers are there?
• n 53
• r 6

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Counting Rule for Permutations

• A third useful counting rule enables us to count
  the number of experimental outcomes when r
  elements are to be selected from a larger set of
  n objects, where the order of selection is
  important.
• The number of permutations of n objects taken r
  at a time is

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Example - Permutation

• If the Florida Lottery requires contestants to
  pick 6 numbers out of 53 in the correct order,
  how many experimental outcomes would there be?

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9.4 Binomial Probability Distribution

• Properties of a Binomial Experiment
• The experiment consists of a sequence of n
  identical trials.
• Two outcomes, success and failure, are possible
  on each trial.
• The probability of a success, denoted by p, does
  not change from trial to trial.
• The trials are independent.

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Example Evans Electronics

• Binomial Probability Distribution
• Evans is concerned about a low retention rate for
  employees. On the basis of past experience,
  management has seen a turnover of 10 of the
  hourly employees annually. Thus, for any hourly
  employee chosen at random, management estimates a
  probability of 0.1 that the person will not be
  with the company next year.
• Choosing 3 hourly employees a random, what is
  the probability that 1 of them will leave the
  company this year?
• Let p .10, n 3, x 1

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Example Evans Electronics

• Success leave within 1 year Fail not
  leave

• Determine the number of experimental outcomes
  involving 1 success in 3 trials (x 1 success in
  n 3 trials).
• Determine the probability of each of the above
  outcomes.

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Example Evans Electronics

• Determine the number of experimental outcomes
  involving 1 success in 3 trials (x 1 success in
  n 3 trials).

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Example Evans Electronics

• Determine the probability of each of the above
  outcomes.

Probability of a particular sequence of trial
outcomes
With x successes in n trials
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Binomial Probability Distribution

• Binomial Probability Function
• where
• f(x) the probability of x successes in n
  trials (x 1)
• n the number of trials (3)
• p the probability of success on any one
  trial (.10)

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Binomial Probability Distribution

• Binomial Probability Function

Step 1
Step 2
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Example Evans Electronics

• Using the Binomial Probability Function
• (3)(0.081)
• .243

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Example Evans Electronics

• Using a Tree Diagram

Second Worker
Third Worker
Value of x
First Worker
Probab.
L (.1)
.0010
3
Leaves (.1)
2
.0090
S (.9)
Leaves (.1)
L (.1)
.0090
2
Stays (.9)
1
.0810
S (.9)
L (.1)
2
.0090
Leaves (.1)
1
.0810
S (.9)
L (.1)
Stays (.9)
1
.0810
Stays (.9)
0
.7290
S (.9)
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Binomial Probability Distribution

• Expected Value
• E(x) ? np
• Evans Electronics
• E(x) ? 3(.1) .3 employees out of 3

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Binomial Probability Distribution

• Seventy percent of the students applying to a
  university are accepted.
• What is the probability that among the next 18
  applicants exactly 10 will be accepted?

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Solution 1
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Binomial Probability Distribution

• Seventy percent of the students applying to a
  university are accepted.
• What is the probability that among the next 18
  applicants exactly 10 will be accepted?
• Determine the expected number of acceptances if n
  18.

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Solution 2
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Binomial Probability Distribution

• Seventy percent of the students applying to a
  university are accepted.
• What is the probability that among the next 18
  applicants exactly 10 will be accepted?
• Determine the expected number of acceptances if n
  18.
• What is the probability that among the next 5
  applicants no more than 3 are accepted?

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Solution 3
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Now You Try, pg. 547, 1, 6
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End of Chapter 8