Download chapter 4 lecture notes and exercises

Math 211
Introduction to Statistics
Chapter 5 Probability and Some Probability Distributions
Key words: Sample space, sample point, tree diagram, events, complement, union and
intersection of an event, mutually exclusive events; Counting techniques: multiplication rule,
permutation, combination, probability of an event; Additive rules; Conditional probability,
independent events, multiplicative rules, Bayes’ rule.
SAMPLE SPACE
Sample Space. The set of all possible outcomes of a statistical experiment is called a Sample
space. It is represented by the symbol S.
Sample Point. Each outcome in a sample space is called an element or a sample point of the
sample space.
Example 1. (a) Consider the experiment of tossing a coin. The sample space S of possible
outcomes may be written as
S= H , T  .
(b) Consider the experiment of flipping a die. Then the elements of the sample
space S is listed as
S= 1, 2,3, 4,5,6 .
(c) Now consider the experiment of tossing a die and then a coin once. The resultant
sample space can be obtained using TREE DIAGRAM. That is,
Therefore the sample sample S is
S= 1H ,1T , 2H , 2T ,3H ,3T , 4H , 4T ,5H ,5T ,6 H ,6T  .
Sonuc Zorlu
1
Lecture Notes
Math 211
Introduction to Statistics
Event. An event is a subset of a sample space.
For example A  4H ,6T  is an event defined on S. One can define 212 events on S.
Empty set  , is an impossible event and S is a sure event. Any subset of S is represented by
capital letters such as A, B, C…
The Complement of an Event. The complement of an event A with respect to S is the subset of
all elements of S that are not in A. The complement of A is denoted by the symbol A ' or Ac .
The Intersection of Events. The intersection of two events A and B, denoted by the symbol
A  B is the event containing all elements that are common to A and B.
Mutually Exclusive Events. Two events A and B are mutually exclusive or disjoint if
A  B   , that is if A and B have no common elements in common.
The Union of Events. The union of two events A and B, denoted by the symbol A  B is the
event containing all elements that belong to A or B or both.
Important Notes. The following results may easily be verified by means of Venn diagrams.
(1) A  
(2) A   A
(3) A  A '  
(4) A  A '  S
(5) S '  
(6)  '  S
(7)  A ' '  A
(8)
(9)
 A  B  '   A ' B ' 1st De Morgan Rule
 A  B  '  A ' B ' 2nd De Morgan Rule
Example 2. If S  0,1, 2,3, 4,5,6,7,8,9 and A  0, 2, 4,6,8 , B  1,3,5,7,9 , C  2,3, 4,5
and D  1,6,7 , list the elements of the sets corresponding to the following events:
(a) A  C  S
(b) A  B   ( A and B are mutually exclusive events)
(c) C '  0,1,6,7,8,9
(d)
 C ' D   B  0,1, 6, 7,8,9  1, 6, 7  1,3,5, 7,9
 1, 6, 7  1,3,5, 7,9
 1, 7
(e)  S  C  '  C '  0,1,6,7,8,9
A  C  D '  2, 4  0, 2,3, 4,5,8,9
(f)
 2, 4
 AC
Sonuc Zorlu
2
Lecture Notes
Math 211
Introduction to Statistics
COUNTING SAMPLE POINTS
Multiplication Rule. If an operation can be performed in n1 ways, and if for each of these a
second operation can be performed in n2 ways, then the two operations can be performed
together in n1n2 ways.
Example 1. How many breakfasts consisting of a drink and a sandwich are possible if we can
select from 3 drinks and 4 kinds of sandwiches?
Solution : n1  3 and n2  4 , therefore there are n1xn2  3x4  12 different ways to choose a
breakfast.
Example 2. A certain shoe comes in 5 different styles with each style available in 4 distinct
colors. If the store wishes to display pairs of these shoes showing all of its various styles and
colors, how many different pairs would the store have on display?
Solution : 5  4  20 different pairs are available.
Example 3. In how many ways can a true-false test consisting of 9 questions be answered?
Solution : There are 2  2  2  2  2  2  2  2  2  29 different answers for this test.
Example 4. How many even three-digit numbers can be formed fro the digits 1,2,5,6, and 9 if
(a) each digit can be used only once?
(b) each digit can be repeated?
Solution :
(a) Since the number must be even we have only 2 choices for the units position.
For each of these there are 4 choices for hundreds position and 3 choices for tens positions.
Therefore we can form a total of 2  4  3  24 even three-digit numbers.
(b) Similarly for the units position there are 2 choices, since the numbers can be repeated we
have 4 choices for the hundreds and tens positions. Hence, totally there are 2  4  4  32 even
three-digit numbers.
Permutation . A permutation is an arrangement of all or part of a set of objects.
The number of permutations of n distinct objects is n! .
Theorem . The number of permutations of n distinct objects taken r at a time is
n!
n Pr 
 n  r !
Sonuc Zorlu
3
Lecture Notes
Math 211
Introduction to Statistics
Example 5 . In how many ways can a Society schedule 3 speakers for 3 different meetings if
they are available on any of 5 possible dates?
Solution : The total number of possible schedules is
5!
 5.4.3  60 .
5 P3 
 5  3 !
Theorem . The number of permutations of n distinct objects arranged in a circle is  n  1! .
Theorem . The number of distinct permutations of n things of which n1 are of one kind, n2 of
a second kind,…, nk of a k th kind is
n!
n1 !n2 !...nk !
Example 6. How many different ways can 3 red, 4 yellow, and 2 blue bulbs be arranged in a
string of Christmas tree lights with 9 sockets?
Solution : The total number of distinct arrangements is
9!
 1260 .
3!4!2!
Example 7. Suppose that 10 employees are to be divided among three jobs with 3 employees
going to job I, 4 to job II, and 3 to job III. In how many ways can the job assignment be made?
Solution.
10!
=2200
3!4!3!
Theorem. The number of combinations of n distinct objects taken r at a time is
n
n!
n Cr    
 r  (n  r )!r !
Example 8. From 4 mathematicians and 6 computer scientists, find the number of committees
that can be formed consisting of 2 mathematicians and 4 computer scientists.
 4  6
4! 6! 4.3.2!6.5.4!
Solution.   .   

 3.6.5  90
2!2.2.4!
 2   4  2!2! 2!4!
Sonuc Zorlu
4
Lecture Notes
Math 211
Introduction to Statistics
Probability of an Event
Definition. Suppose that an experiment has associated with it a sample space S. A probability is
a numerically valued function that assigns a number P  A to every event A so that the
following axioms hold:
(1) P( A)  0
(2) P( S )  1
(3) If A1 , A2 ,..., is a sequence of mutually exclusive events (i.e. Ai  Aj   for any
i  j ), then
  
P  Ai    P( Ai )
 i 1  i 1
Example 1. A coin is tossed twice. What is the probability that at least one head occurs?
The sample space for this experiment is S  HH , HT , TH , TT  . If the coin is balanced,
each of these outcomes would be equally likely to occur. Therefore we assign a probability w to
each sample point. Then 4w=1 or w=1/4.If A represents the event of at least one head
occurring, then
1 1 1 3
A = {HH , HT , TH } and P( A)= + + = .
4 4 4 4
Theorem . If an experiment can result in any one of N different equally likely outcomes, and if
exactly n of these outcomes correspond to event A , then the probability of event A is
n
P( A) =
N
Additive Rules
Theorem. If A and B are two events, then
P (A È B) = P( A) + P( B) - P (A Ç B)
Corollary. If A and B are mutually exclusive events, then
P (A È B) = P( A) + P( B)
Corollary. If A1 , A2 , A3 ,..., An are mutually exclusive events, then
P (A1 È A2 È ... È An ) = P( A1 ) + P( A2 ) + ... + P( An )
Theorem. For three events A , B and C ,
P (A È B È C ) = P( A) + P( B) + P (C )- P (A Ç B)- P (A Ç C )- P (B Ç C )+ P (A Ç B Ç C ).
Sonuc Zorlu
5
Lecture Notes
Math 211
Introduction to Statistics
Example 1. What is the probability of getting a total 7 or 11 when a pair of dice are tossed?
Solution. Let A : event that 7 occurs ; B : event that 11 occurs , therefore
1 1
2
P (A È B ) = P( A) + P( B) = +
=
6 18 9
Theorem. If A and A¢ are complementary events, then
P( A) + P( A¢) = 1
Example 2. In a random experiment it is known that P( A)  0.35, P( A  B)  0.19 ,
and P( A  B )  0.15. Calculate P( B) .
Exercises
Exercise 1. A pair of dice is tossed. Find the probability of getting
(a) a total of 8
(b) at most a total of 5.
Exercise 2. Two cards are drawn in succession from a deck without replacement. What is the
probability that both cards are greater than 3 and less than 9?
Exercise 3. If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems,
and a dictionary, what is the probability that
(a) the dictionary is selected?
(b) 2 novels and 1 book of poems are selected?
Exercise 4. If 2 of the 10 employees are female and 8 male, what is the probability that exactly
one female gets selected among the three?
Exercise 5. A package of 6 light bulbs contain 2 defective bulbs. If 3 bulbs are selected for use,
find the probability that none is defective.
Exercise 6. How many four-digit serial numbers can be formed if no digit is to be repeated
within any one number?
Exercise 7. Seven applicants have applied for two jobs. How many ways can the jobs be filled if
(a) the first person chosen receives a higher salary than the second?
(b) There are no differences between the jobs?
Exercise 8. From a group of 4 men and 5 women, how many committees of size 3 are possible
(a) with no restrictions?
(b) with 1 man and 2 women?
(c) with 2 men and 1 woman if a certain man must be on the committee?
Exercise 9. In how many ways can the letters of the word ‘KRAKATOA’ be arranged?
Sonuc Zorlu
6
Lecture Notes
Math 211
Introduction to Statistics
Random Variables and Probability Distributions
In statistics we deal with random variables- variables whose observed value is determined by
chance. Random variables usually fall into one of two categories: discrete or continuous
Random Variable. A random variable (r.v.) is a function that associates a real number with each
element in the sample space. Random variables will be denoted by uppercase letters and their
observed numerical values by lowercase letters.
Discrete Random Variable. A random variable is discrete if it can assume at most a finite or a
countably infinite number of possible values.
Example 1. Two balls are drawn in succession without replacement from an urn containing 4 red
and 3 black balls. The possible outcomes and values y of the random variable Y where Y is the
number of red balls are:
Sample Space y
RR
2
RB
1
BR
1
BB
0
Continuous Random Variable. A random variable is continuous if it can assume any value in
some interval or intervals of real numbers and the probability that it assumes any specific value
is 0.
Discrete Probability Distributions
Definition. The set of ordered pairs  x, f  x   is a probability function, probability mass
function or probability distribution of the discrete random variable X if,
(1) f  x   0
(2)
 f  x  1
x
(3) P  X  x   f  x  .
Example 1. A committee of size 5 is to be selected at random from 3 chemists and 5
mathematicians. Find the probability distribution (p.d.) for the number of chemists on the
committee.
Let X be the number of chemists on the committee. Then x : 0,1, 2,3 .
Sonuc Zorlu
7
Lecture Notes
Math 211
Introduction to Statistics
 3  5 
  
0 5
1
;
P  X  0   f  0      
56
8
 
 5
 3  5 
  
1 4
15
P  X  1  f 1     
56
8
 
5
 3  5 
 3  5 
  
  
2  3  30
3 2
10

;
P  X  2  f  2 

P  X  3  f  3     
56
56
8
8
 
 
 5
5
Therefore the probability distribution of X is
x
f  x
0
1
56
1
15
56
2
30
56
3
10
56
Exercise 1. Among 10 applicants for an open position 6 are female and are males. Suppose 3
applicants are randomly selected from the applicant pool for final interviews. Find the
probability distribution for X , which is the number of female applicants among the final three.
Exercise 2. Let w be a random variable giving the number of heads minus the number of tails
in three tosses of a coin.
(a) List the elements of the sample space
(b) Assign a value w of W to each sample points.
(c) Find the probability distribution of the random variable W assuming that the coin is
biased so that a head is twice as likely to occur as a tail.
Cumulative Distribution. The cumulative distribution F  x  of a discrete random variable X
with probability distribution f  x  is
F  x   P  X  x    f t  ,
for   x   .
tx
Example 2. Find the cumulative distribution of the number of red balls in example 1. Using
10
F  x  , show that f  3  .
56
F  0  f  0 
Sonuc Zorlu
1
;
56
8
Lecture Notes
Math 211
Introduction to Statistics
F 1  f  0   f 1 
16
56
46
;
56
F  3  f  0  f 1  f  2  f  3  1 .
F  2   f  0   f 1  f  2  
Hence,
0,
1

 56
 16
F  x  
 56
 46
 56

1
46 10
 .
Now, f  3  F  3  F  2   1 
56 56
if x  0
if 0  x  1
if 1  x  2
if 2  x  3
if x  3
Continuous Probability Distributions
Definition. (Probability Density Function) The function f  x  is a probability density
function for the continuous random variable X , defined over the set of real numbers , if
(1) f  x   0,
for all x 

(2)
 f  x  dx  1

b
(3) P  a  X  b    f  x  dx .
a
Note that for a continuous random variable X ,
a
P  X  a    f  x  dx  0 .
a
Cumulative Distribution. The cumulative distribution F  x  of a continuous random variable
X with density function f  x  is
F  x  P  X  x 
x
 f t dt
for    x   .

Sonuc Zorlu
9
Lecture Notes
Math 211
Introduction to Statistics
Example 1. Suppose that a random variable X has a probability density function given by
kx 1  x  0  x  1
f  x  
elsewhere
0
(a) Find the value of k that makes this a probability function.
1
 kx 1  x  dx  1  k  6
0
(b) Find P  0.4  X  1
  0.4 2  0.4 3 
 6 x 1  x  dx  1  6  2  3   0.332
0.4


(c) Find F  x   P  X  x  and sketch the graph of this function.
1
if x  0
0,
 2
F ( x)  3 x  2 x 3 , if 0  x  1
1,
if
x 1

Some Useful Probability Distributions
The observations generated by different statistical experiments have the same general type of
behavior. Random variables associated with these experiments can be described by essentially
the same probability distribution and therefore can be represented by a single formula. The
followings are the probability distributions that will be covered in this chapter:



Binomial Distribution
Poisson Distribution and Poisson Process
Normal Distributions
The Binomial Distribution
Perhaps the most commonly used discrete probability distribution is the binomial distribution.
An experiment which follows a binomial distribution will satisfy the following requirements
(think of repeatedly flipping a coin as you read these):
1. The experiment consists of n identical trials, where n is fixed in advance.
2. Each trial has two possible outcomes, S or F, which we denote ``success'' and ``failure''
and code as 1 and 0, respectively.
3. The trials are independent, so the outcome of one trial has no effect on the outcome of
another.
4. The probability of success, p is constant from one trial to another.
Sonuc Zorlu
10
Lecture Notes
Math 211
Introduction to Statistics
The random variable X of a binomial distribution counts the number of successes in n trials. The
probability that X is a certain value x is given by the formula
n
n x
P  X  x   b  x, n, p     p x 1  p 
 x
n
where 0  p  1 and x  0,1, 2,...., n . Recall that the quantity   , ``n choose x,'' above is
 x
We could use the formulas previously given to compute the mean and variance of X. However,
for the binomial distribution these will always be equal to
E  X     np and Var  X    2  npq
NOTE:\ A particularly important example of the use of the binomial distribution is when sampling
with replacement (this implies that p is constant).
Example 1. Suppose we have 10 balls in a bowl, 3 of the balls are red and 7 of them are blue.
Define success S as drawing a red ball. If we sample with replacement, P(S)=0.3 for every trial.
Let's say n=20, then X b  x, 20,0.3 and we can figure out any probability we want. For
example,
The mean and variance are
Example 2. The probability that a patient recovers from a rare blood disease is 0.4. If 15 people
are known to have contracted this disease, what is the probability that
(a) at least 10 survive?
(b) from 3 to 7 survive?
(c) exactly 5 survive?
Solution. Probability of success= p  0.4 , and the probability of failure= q  0.6 .
n  15 and X : no. of surviving patients
(a) P  X  10  1  P  X  9  1  B 9;15,0.4  1-0.9662=0.0338
(b) P  3  X  7   P  4  X  6  b  4;15,0.4  b  5;15,0.4  b  6;15,0.4  0.509
15 
5
155
 0.186
(c) P  X  5  b  5;15, 0.4      0.4   0.6 
5 
Sonuc Zorlu
11
Lecture Notes
Math 211
Introduction to Statistics
Example 3. A traffic control engineer reports that 75% of the vehicles passing through a
checkpoint are from within the state. What is the probability that fewer than 2 of the next 9
vehicles are from out of the state?
Solution. Probability of success= p  0.25 , and the probability of failure= q  1  0.25  0.75 .
n  9 and X : no. of vehicles passing through the checkpoint
P  X  2   P  X  1  b  0;9, 0.25   b 1;9, 0.25 
9
9
0
9
1
8
    0.25   0.75      0.25   0.75   0.3
0
1 
Example 4. Assuming that 6 in 10 automobile accidents are due mainly to speed violation,
(a) find the probability that among 8 automobile accidents 6 will be due mainly to a
speed violation.
(b) Find the mean and variance of the number of automobile accidents for 8 automobile
accidents.
Solution. Probability of success= p  6 /10 , and the probability of failure= q  1  0.25  0.75 .
and X : no. of automobile accidents
n 8
6
8 6
8  6  
6
(a) P  X  6   b  6;8, 6 /10      1    0.2090
 6   10   10 
(b) The mean of the number of automobile accidents is   E  X   np  8 *
The variance of the no. of auto. accidents is  2  Var  X   npq  8 *
6
 4.8 .
10
6 4
*  1.92 .
10 10
The Poisson Distribution
The Poisson distribution is most commonly used to model the number of random occurrences of
some phenomenon in a specified unit of space or time. For example,



The number of phone calls received by a telephone operator in a 10-minute period.
The number of flaws in a bolt of fabric.
The number of typos per page made by a secretary.
For a Poisson random variable, the probability that X is some value x is given by the formula
P  X  x   f  x;  t  
e t   t 
x!
x
x  0,1, 2,...
where  is the average number of occurrences per unit time or region denoted by t . For the
Poisson distribution,
Sonuc Zorlu
12
Lecture Notes
Math 211
Introduction to Statistics
E  X    t and Var  X    t .
Example 1. The number of false fire alarms in a suburb of Houston averages 2.1 per day.
Assuming that a Poisson distribution is appropriate, the probability that 4 false alarms will occur
on a given day is given by
Example 2. During a laboratory experiment the average number of radioactive particles passing
through a counter in 1 millisecond is 4. What is the probability that 6 particles enter the counter
in a given millisecond?
Example 3. A secretary makes 2 errors per page, on average. What is the probability that on the
next page he or she will make
(a) 4 or more errors?
(b) no errors?
Example 4. The number of customers arriving per hour at a certain automobile service facility is
assumed to follow a Poisson distribution with   7 .
(a) Compute the probability that more than 10 customers will arrive in a 3-hour period.
(b) What is the mean number of arrivals during a 4-hour period?
Example 5. A restaurant chef prepares tossed salad containing, on average, 5 vegetables. Find
the probability that the salad contains more than 5 vegetables
(a) on a given day
(b) on 3 of the next 4 days
(c) for the first time in April on April 5.
Poisson Approximation to Binomial Distribution
Theorem. Let X be a binomial random variable with probability distribution b  x; n, p  . When
n  , p  0 and   np remains constant,
b  x; n, p   p  x;   .
Example 1. The probability that a person will die from a certain respiratory infection is 0.002.
Find the probability that fewer than 5 of the next 2000 so infected will die.
X : number of infected people who will die
Given n  2000, p  0.002 ,   np  2000 * 0.002  4 ,
4
P  X  5   P  X  4    b  x; 2000, 0.002 
n l arg e , p small 4
x 1
Sonuc Zorlu
13

 p  x; 4   P  4; 4 
by table A 2

0.6288.
x 1
Lecture Notes
Math 211
Introduction to Statistics
Example 2. It is known that 5% of the books bound by a certain bindery have defective
bindings. Find the probability that 2 of 100 books bound by this bindery will have defective
bindings using
(a) the formula for the binomial distribution
(b) the Poisson approximation to the Binomial distribution.
Solution. (a) Given p  0.05, n  100 , and X , number of defective bindings,
100 
2
98
P  X  2   b  2;100, 0.05  
  0.05  0.95  0.081
2 
e5 52
(b) P  X  2   b  2;100, 0.05  p  2;5 
 0.084 where   np  100 * 0.05  5 .
2!
The Normal Distribution
The most important continuous probability distribution in the entire field of statistics is the
normal distribution. Normal distributions are a family of distributions that have the same general
shape. They are symmetric with scores more concentrated in the middle than in the tails. Normal
distributions are sometimes described as bell shaped which are shown below. Notice that they
differ in how spread out they are. The area under each curve is the same. The height of a normal
distribution can be specified mathematically in terms of two parameters: the mean (μ) and the
standard deviation (σ).
Definition. The density function of the normal random variable X , with mean  and variance
 2 , is
2
1
1/ 2   x    /  
N  x;  ,   
e
,   x  
2
where   3.14159.... and e  2.71828 .
Sonuc Zorlu
14
Lecture Notes
Math 211
Introduction to Statistics
Standard normal distribution
The standard normal distribution is a normal distribution with a mean of 0 and a standard
deviation of 1. Normal distributions can be transformed to standard normal distributions by the
formula:
where X is a score from the original normal distribution, μ is the mean of the original normal
distribution, and σ is the standard deviation of original normal distribution. The standard normal
distribution is sometimes called the z distribution. A z score always reflects the number of
standard deviations above or below the mean a particular score is. For instance, if a person
scored a 70 on a test with a mean of 50 and a standard deviation of 10, then they scored 2
standard deviations above the mean. Converting the test scores to z scores, an X of 70 would be:
So, a z score of 2 means the original score was 2 standard deviations above the mean. Note that
the z distribution will only be a normal distribution if the original distribution (X) is normal.
Note : The following figures give us the areas to the right of some z  value, to the left of some
z  value and between two z  values.
Sonuc Zorlu
15
Lecture Notes
Math 211
Introduction to Statistics
Areas under portions of the standard normal distribution are shown to the right. About .68 (.34 +
.34) of the distribution is between -1 and 1 while about .96 of the distribution is between -2 and
2.
Example 1. Given a standard normal distribution, find the area under the curve that lies
(a) to the right of z  1.84
(b) between z  1.97 and z  0.86
Solution. (a) P  Z  1.84   1  P  Z  1.84 
by table A3

1  0.9671  0.0329 .
(b) P  1.97  Z  0.86  P  Z  0.86  P  Z  1.97 
by table A3

0.8051  0.0244  0.7807
Example 2. Given a normal distribution with   50 and   10 , find the probability that
X assumes a value between 45 and 62.
Solution.
62  50 
 45  50
P  45  X  62   P 
Z
  P  0.5  Z  1.2   table(1.2)  table(0.5)
10 
 10
= 0.8849-0.3088  0.5764
Example 3. Given a standard normal distribution, find the value of k such that
(a) P  Z  k   0.0427
(b) P  Z  k   0.2946
(c) P  0.93  Z  k   0.7235
Solution. (a) P  Z  k   0.0427  table(k )  0.0427  k  1.72
(b)
P  Z  k   0.2946  P  Z  k   1  P  Z  k   0.2946
 P  Z  k   1  0.2946  0.7054
 table(k )  0.7054  k  0.54
(c) P  0.93  Z  k   0.7235  table(k )  table(0.93)  0.7235
table(k )  0.7235  0.1762  0.8997
 k  1.28
Sonuc Zorlu
16
Lecture Notes
Math 211
Introduction to Statistics
Example 4. Given the normally distributed random variable X with  X  18 and  X2  3
(a) Compute P  X  13.74
(b) Compute x satisfying P  x  X  18  0.4332 .
Applications of the normal Distribution
Example 1. A certain machine makes electrical resistors having a mean resistance of 40 ohms
and a standard deviation of 2 ohms. Assuming that the resistamce follows a normal distribution
and can be measured to any degree of accuracy, what percenatge of resistors wil have a
resistance exceeding 43 ohms?
Solution. Let X be the normal random variable, given   40ohms,   2ohms ,
 X   43  40 
P  X  43  P 

  P  Z  1.5   1  P  Z  1.5 
2 
 
 1  table(1.5)  1  0.9332  0.0668  6.68%
Example 2. The average grade for a exam is 74 and the standard deviation is 7. Assuming that
the grades follow a normal distribution, what is the probability that a student will get grade of at
least 60?
Normal Approximation to Binomial Distribution
Theorem. If X is a binomial random variable with mean   np and  2  npq , then the
limiting form of distribution of
X  np
Z  bin
as n  
npq
is the standard normal distribution N  0,1 .
Note. We use the normal approximation to binomial distribution whenever p is not close to 0
and 1. If both np and nq are greater than or equal to 5, the approximation will be good.
Example 1. The probability that a patient recovers from a blood disease is 0.4. If 100 people are
known to have contracted this disease what is the probability that less than 30 survive?
Solution. Let X be the number of surviving people from blood disease.
Given n  100 and p  0.40 ,   np  100 * 0.40  40 ,   100 * 0.40 * 0.60  0.24 ,
Sonuc Zorlu
17
Lecture Notes
Math 211
Introduction to Statistics
 X  np 29.5  40 
P  X bin  30   P  X nor  29.5   P 

  P  Z  2.14   table(2.14)  0.0162
 npq
24 

Example 2. A coin is tossed 400 times, use the normal approximation to binomial to find the
probability of obtaining
(a) between 185 and 210 heads inclusive
(b) exactly 205 heads
(c) less than 176 or more than 227 heads
Example 3. A ablanced die is rolled 180 times. Let be the number of cases when die shows the
number 4 on its top face.
(a) Find  X
(b) Find  X2
(c) Use normal approximation to binomial to approximate P  35  X  40 .
Exercises
Exercise1. If scores are normally distributed with a mean of 30 and a standard deviation of
5, what percent of the scores is: (a) greater than 30? (b) greater than 37? (c) between 28 and
34?
Exercise 2. Assume a normal distribution with a mean of 90 and a standard deviation of 7.
What limits would include the middle 65% of the cases?
Exercise 3. If is the standard normal random variable,
(a) Calculate P  1.3  Z  1.37 
(b) If P  a  Z  1.12  0.6845 , find the value of a .
Exercise 4. A research scientists reports that mice will live an average of 40 months when
their diets are sharply restricted and enriched with vitamins and proteins. Assuming that the
lifetimes of such mice are normally distributed with a standard deviation of 6.3 months, find
the probability that a given mouse will live
(a) more than 32 months
(b) less than 28 months
(c) between 37 and 49 months
Sonuc Zorlu
18
Lecture Notes