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7. Logic, Sets, and Counting
7.3 Basic Counting Principles
Chemistry class – 13 males, 15 females
Set M, set F, M  F , M  F , nM  F , nM  F 
Algebra – 22 Math, 16 Physics, 7 double majors
Set M, set P, M  P , M  P , nM  P 
n M  P  
Addition Principle (for Counting)
For any two sets A and B,
n A  B   n A   n B   n A  B 
If A and B are disjoint, then
n A  B   n A   n B 
According to a survey, 750 businesses offer health
insurance to their employees, 640 offer dental
insurance, and 280 offer both. How many offer
health or dental insurance?
Venn diagrams
A city has two daily newspapers. A survey of 100
residents shows that 35 subscribe to the Sentinel, 60
to the Journal, and 20 to both.
A. How many subscribe to only the Sentinel?
B. How many subscribe to only the Journal?
C. How many subscribe to neither paper?
D. Organize this information in a table.
Multiplication Principle (for Counting)
A store stocks windbreaker jackets in S, M, L, and
XL, and all are available in blue and red. What are
the combined choices, and how many combined
choices are there?
If two operations O1 and O2 are performed in order,
with N1 possible outcomes of the first and N 2
possible outcomes of the second, then there are
N1  N 2 possible outcomes of the first followed by
the second.
How many ways can 3 letters and 3 numbers
appear on a license plate?
4 letters and two numbers?
Suppose a university screening test is to consist
of 5 questions, and the generating computer
stores 5 comparable first questions, 8 second, 6
third, 5 fourth, and 10 fifth. How many
different 5-question tests can be generated?
How many 3-letter code words can be generated
using the first 8 letters of the alphabet if:
A. No letter is repeated?
B. Letters can be repeated?
C. Adjacent letters cannot be alike?
7.4 Permutations and Combinations
Factorials
For n, a natural number,
n! n  n  1  n  2     2 1
0! 1
n! n  n  1!
5!=
7!

6!
8!

5!
52!

5!47!
Permutations
A permutation of a set of distinct objects is an
arrangement of the objects in a specific order
without repetition.
How many ways can 5 pictures be arranged on a
wall?
Number of permutations of n objects
The number of permutations of n distinct
objects without repetition, designated n Pn , is
n
Pn  nn  121  n!
Permutations of n objects taken r at a time
How many ordered arrangements of 3 pictures
can be formed from the 5 pictures above?
The number of permutations of n distinct
objects taken r at a time, is given by
Pr  nn  1n  r  1
or
n!
P

n r
n  r !
n
Given the set A, B, C, how many permutations
are there of this set taken 2 at a time?
A. Using a tree diagram?
B. Multiplication principle
C. Two formulas for n Pr
Find the number of permutations of 13 objects
taken 8 at a time.
P 
13 8
Combinations
A combination of n objects taken r at a time
without repetition is an r-element subset of the
set of n objects. The arrangement of the
elements in the subset does not matter.
Symbolism
n
Cr
n
 
r
Combinations of n objects taken r at a time
The number of combinations of n objects taken
r at a time without repetition is given by
n
Cr
n
= 
r
Pn ,r

r!

n!
r!n  r !
From a committee of 10 people,
A. How many ways can we choose a chair,
vice-chair, and a secretary?
B. How many ways can we choose a
subcommittee of 3 people?
Find the number of combinations of 13 objects
taken 8 at a time.
n Cr 
Permutation  order is vital
Combination  order is irrelevant
How many 5-card hands will have 3 aces and
2 kings?
Serial numbers for a product are to be made
using 2 letters followed by 3 numbers. If letters
are selected from the first 8 letters of the
alphabet with no repeats, and numbers are to be
taken from the 10 digits (0-9), with no repeats,
how many serial numbers are possible?
A company has 7 senior officers and 5 junior
officers. How many ways can a 4-officer
committee be formed composed of:
A. any four officers
B. 4 senior officers
C. 3 seniors and a junior
D. 2 seniors and 2 juniors
E. at least 2 seniors
From a standard 52-card deck, how many 3-card
hands have all cards from the same suit?
n
Cr 
8. Probability
8.1 Sample Spaces, Events, and Probability
Theoretical approach, empirical approach
If we formulate a set S of outcomes (events) in
such a way that each trial of an experiment has
only one possible outcome out of a set, we call
that set a sample space and each outcome, a
simple outcome or simple event. An event E
is defined to be any subset of S (including the
empty set and the entire set). E is a simple
event if it includes only one element, and a
compound event if more than one.
Simple roulette wheel – 18 spaces S  1,2,3,,17,18
Desired outcome: divisible by 4 E  4,8,12,16
Outcome is a prime number S  2,3,5,7,11,13,17
Outcome is the square of 4 S  16
Sample spaces
A nickel and a dime are flipped. Sample space?
Possible outcomes
Number of heads
Match or Don’t match
Rolling two dice
1
2
3
4
5
6
1
1,1
2,1
3,1
4,1
5,1
6,1
2
3
4
5
6
1,2 1,3 1,4 1,5 1,6
2,2 2,3 2,4 2,5 2,6
3,2 3,3 3,4 3,5 3,6
4,2 4,3 4,4 4,5 4,6
5,2  5,3 5,4  5,5 5,6
6,2 6,3 6,4 6,5 6,6 
Some events:
Roll a 7 6,1, 5,24,33,42,51,6
Roll 11
Sum less than 4
Sum of 12
Probability of an event
Acceptable probability assignment
Given a sample space
S  e1 , e2 ,, en 
we assign a real number, denoted Pei , called
the probability of event ei .
1. Each probability is between 0 and 1
0  Pei   1
2. Sum of all probabilities is 1.
Pe1   Pe2     Pen   1
Tossing a coin
S  H , T 
P H   1
2
Theoretical approach
Empirical approach
PT   1
2
Probability of an event E
# favorable outcomes
# possible outcomes
Given an acceptable probability assignment for
simple events in sample space S, the
probability of an arbitrary event E, denoted
P E , is as follows:
a. If E is the empty set, then P E   0 .
b. If E is a simple event, P E  has already
been assigned.
c. If E is a compound event, then P E  is
the sum of the probabilities of the simple
events in E.
d. If E=S, P E   1.
A nickel and a dime are flipped.
Sample space S  HH , HT , TH , TT 
Event
ei
Pei 
HH
HT
TH
TT
1
1
1
1
4
4
4
4
Probability of one head and one tail?
Probability of at least one head?
Probability of at least one head or at least one tail?
Probability of 3 heads?
To find probability of an event E:
1. Setup an appropriate sample space S for the
experiment.
2. Assign acceptable probabilities to each
simple event in S.
3. To obtain the probability for an arbitrary
event E, add the probabilities of the simple
events in E.
Theoretical approach: We use assumptions and
deductive reasoning to assign probabilities.
Empirical approach: We assign probabilities
based on results of experiments.
Frequency f E 
Relative frequency f  E 
n
Empirical Probability P E   lim f  E 
n 
n
Empirical Probability approximation
P E  
frequency of occurrence of E f  E 

n
total number of trials
Equally likely assumption
In a sample space
S  e1 , e2 ,, en 
with n elements, we assume that each simple
event ei is as likely to occur as any other. Then
P E   1
n
Rolling a prime number with die
P E  
Probability of an Arbitrary Event under an
Equally Likely Assumption
If we assume that each simple event in simple
space S is as likely to occur as any other, then
the probability of an arbitrary event E in S is
given by:
number of elements in E nE 
P E  

n S 
number of elements in S
Consider rolling two dice,
Roll 7
PE1  
E16,1, 5,24,33,42,51,6
Roll 11
PE2  
Sum less than 4
PE3  
Sum of 12 PE4  
In drawing 5 cards from a 52 card deck without
replacement, what is the probability of getting 5
spades?
P E  
The board of regents is made up of 12 men and
16 women. If a committee of 6 is chosen at
random, what is the probability that it will
contain 3 men and 3 women?
P E  
8.2 Union, Intersection, and Complement of
Events; Odds
A  B  e  S e  A or e  B
A  B  e  S e  A and e  B
The event A or B is defined as A  B
The event A and B is defined as A  B .
Let A be the probability of rolling an odd number,
and B be the probability of divisible by 3.
Probability of odd and divisible by 3?
Probability of odd or divisible by 3?
Probability of a Union of Two Events
For any events A and B,
P  A  B   P  A  P  B   P  A  B 
If A and B are disjoint, then
P  A  B   P  A  P  B 
Probability of rolling a 7 or 11?
Probability of rolling less than 5 or doubles?
Probability that a number  500 is exactly
divisible by 3 or 4?
Complement of an Event
Suppose we divide S  e1 , e2 ,, en  into two
disjoint subsets.
E  E '   and E  E '  S .
Then E ' is called the complement of E relative
to S.
Also, PS   P E  E '  P E   P E '  1
P  E   1  P  E '
P  E '  1  P  E 
If the probability of having a boy in a two child
family is 0.75, what is the probability of 2 girls?
A shipment of 45 precision parts, including 9
that are defective, is sent to an assembly plant.
The quality control division selects 10 parts at
random for testing and rejects the entire
shipment of 1 or more in the sample are found
defective. What is the probability that the
shipment will be rejected?
In a group of n people, what is the probability
that at least two people have the same birthday
(same month/day, excluding 29 Feb)
Odds
Odds for E =
P E 
P E 

1  P  E  P  E '
P  E '
Odds against E =
PE 
PE   1
P E   0
Probability and Odds of rolling a 4 with one die:
Probability P4 
Odds
1
6
P E 

P  E '
What are the odds for rolling a 7 in a single roll
of two dice?
If you bet $1 on rolling a 7, what should the
house pay if you roll a 7 for the game to be fair?
If the odds for an event E are a b , then the
probability of E is
a
P E  
ab
If in repeated rolls of two fair dice, the odds of
rolling a 5 before a 7 are 2 to 3, then the
probability of rolling a 5 before a 7 is:
Applications to Empirical Probability
Law of Averages
The approximate empirical probability can be
made as close to the actual probability as we
please by making the sample size sufficiently
large.
From a survey of 1,000 people, it was
determined that 500 people had tried a certain
brand of diet soda, 600 had tried the brand of
regular soda, and 200 had tried both. If a
resident is selected at random, what is the
(empirical) probability that:
The resident has tried diet or regular soda?
What are the (empirical) odds for this event?
The resident has tried one but not both? What
are the (empirical) odds against this event?
8.3 Conditional Probability, Intersection, and
Independence
Conditional Probability P A B 
Occurrence of an event, given the occurrence of
another event.
A=adult has lung cancer
B=adult is a heavy smoker
What is the probability of rolling a prime
number?
n A 
P  A 
n S 
What is the probability of the number being
prime if we know an odd number has occurred?
n A  B 
P A B  
n B 
Conditional Probability
For events A and B in arbitrary sample space S,
we define conditional probability of A given B
by
P A  B 
P B   0
P A B  
P B 
A pointer is spun once on a circular spinner with
probabilities below:
ei
1
2
3
4
5
6
Pei  .1
.2
.1
.1
.3
.2
A. What is the probability of landing on a
prime number?
B. What is the probability of landing on a
prime number given it is an odd number?
Suppose that past records in a large city
produced the following probability data on a
driver being in an accident on the last day of a
Memorial Day weekend.
Accident No accident Totals
A
A’
Rain R
.025
.335
.360
No Rain R’ .015
.625
.640
Totals
.040
.960
1.000
A. Find the probability of an accident, rain or no rain.
B. Find the probability of rain, accident or no
accident.
C. Find the probability of accident and rain.
D. Find the probability of accident, given rain.
Intersection of Events: Product Rule
P A B  
P A  B 
P B 
and
P  B A 
P  B  A
P  A
Product Rule
For events A and B with non-zero probabilities in a
sample space S,
P A  B   P APB A  PB P A B 
If 60% of a department store’s customers are
female and 75% of the female customers have
charge accounts at the store, what is the probability
that a customer selected at random is female and
has a charge account?
Probability Trees
Two balls are drawn in succession, without
replacement, from a box containing 3 blue and 2
white balls. What is the probability of drawing a
white ball on the second draw?
Constructing Probability Trees
1. Draw a tree diagram corresponding to all
combined outcomes of the sequence of
experiments.
2. Assign a probability to each branch.
(Probability of event at right end of branch
given the occurrence of the other events
leading to it)
3. Use the results to answer various questions
related to the sequence as a whole.
Computer company A subcontracts circuit board
production 40% to company B, and 60% to
company C. B subs 70% to D and 30% to E.
Completed boards are shipped direct back to A.
1.5%, 1%, and .5% from D, E, and C, respectively
prove defective in first 90 days. What is the
probability of a defective board?
Independent Events
Without replacement
vs.
with replacement
Independence
If A and B are events in sample space S, we say
that A and B are independent iff
P A  B   P AP B 
Otherwise, A and B are said to be dependent.
Thm 1
If A and B are independent events with nonzero
probabilities, in a sample space S, then
P A B   P A and PB A  PB 
P A  B 
P  B  A
Since P A B  
and PB A 
P B 
P  A
Testing for independence
Consider sample space of two coin tosses.
S  HH , HT , TH , TT  and events
A= head on first toss =HH , HT 
B= head on second toss =HH , TH 
Draw a card from a deck
A. E=drawn card is a spade.
F=drawn card is a face card.
B. G=drawn card is a club.
H=drawn card is a heart.
A set of events is said to be independent if for
each finite subset E1 , E2 ,, En 
PE1  E2    En   PE1 PE2  PEn 
A space shuttle has 4 independent computer
control systems. If the probability of failure
(during flight) of any one system is 0.001, what is
the probability of the failure of all four systems?
8.4 Bayes’ Formula
Probability of an earlier event, given a later
event.
One urn has 3 blue and 2 white balls, a second urn
has 1 blue and 3 white balls. A single fair die is
rolled, and if 1 or 2 comes up, the ball is drawn
from the first urn, otherwise ball is drawn from
the second urn. If the drawn ball is blue, what is
the probability that it came from the first urn?
M
c
U
a
d
N
b
e
M
f
N
S
V
PM U 
PU M 
Bayes’ Formula
Let U 1,U 2 ,,U n be mutually exclusive events
whose union is sample space S. Let E be an
arbitrary event in S such that P E   0 . Then,
PU1  E 
PU1 E  
P E 
PU1  E 

PU1  E   PU 2  E     PU n  E 
PE U1 PU1 

PE U1 PU1   PE U1 PU1     PE U1 PU1 
product of branch prob to E thru U1
PU1 E  
sum of all branches to E
Tuberculosis screening
A trusted test for TB shows 8% of 1000 in a test
group have TB. A new test indicates TB in 96%
who have it, and in 2% who do not have it. What
is the probability of a random person having it
testing positive? What is the probability of a
person not having it testing positive?
A company produces 1000 refrigerators a week.
Plant A produces 350, plant B produces 250, and
plant C produces 400. Records indicate 5%
from plant A, 3% from plant B, and 7% from
plant C are defective. If a refrigerator is found
to be defective, what is the probability it is from
plant A?
8.5 Random Variable, Probability Distribution,
and Expected Value
A random variable is a function that assigns a
numerical value to each simple event in a
sample space S.
3 coin tosses
TTT
TTH
THT
HTT
THH
HTH
HHT
HHH
p x  where x  0,1,2,3
Probability distribution of the random variable X
Exactly 2 heads occur
E  THH , HTH , HHT 
p2 
n E  3

n S  8
The probability distribution of a random variable
X, denoted by P X  x   p x , satisfies
1. 0  p x   1
2. p x1   p x2     p xn   1
where x1 , x2 ,, xn are the values of X.
Expected value of a Random Variable
E  X   i xi  pi  x 
n
Given the probability distribution for the
random variable X,
xi x1 x 2  xn
pi p1 p2  pn
where pi  p xi , we define the expected value
of X, denoted E  X , by the formula
E  X   x1 p1  x2 p2    xn pn
What is the expected value of the number of
dots facing up on the roll of a single die?
A carton of 20 laptop batteries contains 2 dead
ones. A random sample of 3 is selected from
the 20 and tested. Let X be the random variable
associated with the number of dead batteries
found in a sample.
A. Find the probability distribution of X.
B. Find the expected number of dead batteries
in a sample.
A spinner device is numbered 0 to 5, each
number is equally likely to occur. Any player
who bets $1 on any given number wins $4 (and
gets his bet back) if the pointer lands on the
chosen number, otherwise, the $1 is lost. What
is the expected value of the game?
Suppose you are considering insurance on a
$2000 car video system against theft. The
company charges $225/year, claiming empirical
probability of 0.1 that the stereo will be stolen
some time during the coming year. What is the
expected return to the insurance company if you
take out this policy?
Consider exam scores 85, 73, 82, 65, 95, 85, 73,
75, 85, and 75.
Class average (mean)
E X  
Decision analysis
An outdoor concert featuring a very popular
musical group is scheduled for Sunday
afternoon in a large open stadium. The
promoter, worried about a rainout, hears from a
forecaster the probability of rain is 0.24. If it
does not rain, the promoter will net $100,000, if
it does rain, the promoter will net $10,000. An
insurance company agrees to insure the concert
for $100,000 against rain at a premium of
$20,000. Should the promoter buy the
insurance?
11. Data Description and Probability Distributions
11.2 Measures of Central Tendency
Measures that indicate the approximate center of
a distribution are called measures of central
tendency.
Measures that indicate the amount of scatter
about a central point are called measures of
dispersion.
Mean
The mean of a set of quantitative data is equal
to the sum of all measurements in the data set
divided by the total number of measurements in
the set.
Notation: x =sample mean
 =population mean
n
x
i 1
i
 x1  x2    xn
Mean: ungrouped data
If x1 , x2 ,, xn is a set of measurements, then the
mean is the set of measurements is given by:
n
x
x1  x2    xn
n
n
Use symbol x for sample mean or  for
population mean.
i 1
i

Find the mean for sample measurements 3, 5, 1,
8, 6, 5, 4, and 6.
Mean: grouped data
A data set of n measurements is grouped into k
classes in a frequency table. If xi is the
midpoint of the ith class interval and f i is the ith
class frequency, the mean for the grouped data
n
x f
x1 f1  x2 f 2    xn f n
is given by

n
n
Use symbol x for sample mean or  for
population mean.
i 1
i i
Note that n is the total number of measurements
in all the classes, not the number of classes.
Class interval Midpoint
Frequency Product
299.5 - 349.5
324.5
1
324.5
349.5 - 399.5
374.5
2
749.0
399.5 - 449.5
424.5
5
2122.5
449.5 - 499.5
474.5
10
4745.0
499.5 - 549.5
524.5
21
11014.5
549.5 - 599.5
574.5
20
11490.0
599.5 - 649.5
624.5
19
11865.5
649.5 - 699.5
674.5
11
7419.5
699.5 - 749.5
724.5
7
5071.5
749.5 - 799.5
774.5
4
3098.0
Suppose the annual salaries of seven people in a
small company are $34k, $36k, $36k, $40k,
$48k, $56k, and $156k.
x=
Median
If the number of measurements in a set is odd,
the median is the middle measurement, when
the measurements are arranged in ascending or
descending order.
If the number of measurements in a set is even,
the median is the mean of the two middle
measurements, when the measurements are
arranged in ascending or descending order.
Median for grouped data
The median for grouped data with no classes
of frequency 0 is the number such that the
histogram has the same area to the left of the
median as to the right of the median.
Class interval Frequency
3.5 - 4.5
3
4.5 - 5.5
1
5.5 - 6.5
2
6.5 - 7.5
4
7.5 - 8.5
3
8.5 - 9.5
2
Mode
The mode is the most frequently occurring
measurement in the data set.
4,5,5,5,6,6,7,8,12
1,2,3,3,3,5,6,7,7,7,23
1,3,5,6,7,9,11,15,16
11.3 Measures of Dispersion
Range
The range for a set of ungrouped data is the
difference between the largest and the smallest
values in the data set.
The range for a frequency distribution is the
difference between the upper boundary of the
highest class and the lowest boundary of the
lowest class.
Standard Deviation: Ungrouped data
5.2, 5.3, 5.2, 5,5, 5.3
n
x
x
i 1
n
i

x1  x2    xn
n
n
2

x

x

 i
variance =
i 1
n
n
 x  x 
standard deviation =
i 1
2
i
n
Definitions:
The sample variance s 2 of a set of n sample
measurements x1 , x2 ,, xn with mean x is given by
n
2

x

x

 i
i 1
n 1
If x1 , x2 ,, xn is the whole population with mean  ,
then the population variance  2 is given by
n
 x   
2
i 1
i
n
The sample standard deviation s of a set of n sample
measurements x1 , x2 ,, xn with mean x is given by
n
 x  x 
i 1
2
i
n 1
If x1 , x2 ,, xn is the whole population with mean  ,
then the population standard deviation  is given by
n
2

x



 i
i 1
n
Find the standard deviation for the sample
measurements 1,3,5,4,3
Standard Deviation: grouped data
Suppose a data set of n sample measurementsis
grouped into k classes in a frequency table,
where xi is the midpoint and f i is the frequency
of the ith class interval. Then the sample
standard deviation s for the grouped data is
n
2

x

x

fi
 i
i 1
n 1
k
where i 1 f i = the total number of measurements.
If x1 , x2 ,, xn is the whole population with mean
 , then the population standard deviation  is
given by
n
2

x



fi
 i
i 1
n
Find the standard deviation for each set of
grouped sample data:
11.4 Bernoulli Trials and Binomial Distributions
An experiment for which there are only two
possible outcomes E or E’, is called a Bernoulli
experiment or trial.
Jacob Bernoulli (1654-1705)
Coin flip 7 or not 7 vaccination works or not
P S   p
PF  1  p  q
Probability of rolling a 6 with one die
Suppose a probability experiment is repeated
5x, probability of SSFFS?
Bernoulli Trials
A sequence of experiments is called a sequence
of Bernoulli trials, or a binomial experiment if:
1. Only two outcomes are possible on each
trial.
2. The probability of each success p for each
trial is constant.
3. All trials are independent.
If we roll a die 5x, and define success as rolling
a 1, what is the probability of SFFSS occurring?
Find the probability of the outcome FSSSF.
How many different ways can we roll exactly
three 1’s?
What is the probability of rolling exactly 3 ones?
Probability of x successes in n Bernoulli trials
The probability of exactly x successes in n
independent repeated Bernoulli trials, with the
probability of success of each trial p (and failure
q), is
P(x successes)= n C x p x q n x
What is the probability of rolling exactly two 3’s?
What is the probability of rolling at least two 3’s?
Binomial Formula
a  b 1 
a  b 2 
a  b 3 
a  b 4 
a  b 5 
Binomial Formula
For a natural number n,
a  bn n C0 a n  n C1a n1b n C2 a n2b2   n Cnbn
q  p 3 
Binomial Distribution
3 coin tosses
p x  where x  0,1,2,3
X3
P X 3  x 
Simple Event Probability of
Simple Event x success
in 3 trials
0
TTT=FFF
TTH=FFS
1
THT=FSF
HTT=SFF
THH=FSS
2
HTH=SFS
HHT=SSF
3
HHH=SSS
Binomial Distribution
P X n  x   P( x successes in n trials)
 Cn , x p x q n  x
x  0,1,2,, n
where p is the probability of success and q is the
probability of failure on each trial.
Suppose a fair die is rolled three times; success
is considered a roll of a number divisible by 3.
Probability function
x
0
1
2
3
P x 
Mean and Standard Deviation
Mean:
  np
Standard Deviation:   npq
Compute mean and standard deviation for
previous example
11.5 Normal Distributions
1
  x   2 / 2 2
f x 
e
 2
Normal Curves
1. Normal curves are bell shaped and are
symmetrical with respect to a vertical line.
2. The mean is at the point where the axis of
symmetry intersects the horizontal axis
3. The shape of the normal curve is
completely determined by its mean and
standard deviation.
4. The area under this curve is 1.
5. 68.3% of the area is within 1 SD, 95.4% is
within 2SD, 99.7% is within 3 SD.
A manufacturing process produces light bulbs
with life expectancies that are normally
distributed with a mean of 500 hours and a
standard deviation of 100 hours. What
percentage of bulbs can be expected to last
between 500 hours and 670 hours?
z
x

From all the light bulbs produced, what is the
probability of a light bulb chosen at random
lasting between 380 and 500 hours?
Normal Probability Distribution
1. Pa  x  b   area under normal curve, a to b.
2. P   x     1
3. P x  c   0
What is the probability of a light bulb having a
life of exactly 621 hours?
Approximating a Binomial Distribution with a
Normal Distribution
A credit card company claims that their card is
used by 40% of the people buying gasoline in a
particular city. A random sample of 20 gasoline
purchasers is made.
What is the probability that 6 to 12 people in the
sample use the card?
What is the probability that less than 4 people in
the sample use the card?
Rule of Thumb test
Use of a normal distribution to approximate a
binomial distribution is justified iff the interval
  3 ,   3  lies entirely in the interval from
0 to n.
A company manufactures 50,000 ballpoint pens
each day. The manufacturing process produces
50 defective pens per 1000 on the average. A
random sample of 400 pens is selected from
each day’s production for test. What is the
probability that the sample contains:
A. At least 14 and no more than 25 defective pens?
B. 33 or more defective pens?