Business Statistics
Theory, Problems and Solutions
Probability
Distribution of a random variable is a statement specifying the set of its
possible values together with their respective probabilities. Random variable
is a function which assumes real values on the outcomes of a random experiment.
We shall discuss here mainly four important probability distributions, viz.
Binomial, Poisson, Hypergeometric and Normal.
Binomial,
Poisson and Hypergeometric distributions are Discrete Probability
Distributions, whereas Normal Distribution is a Continuous Probability
Distribution.
|
Probability
function of Binomial distribution is |
|
P(X
= k) = nck.pk.q(n − k) |
|
Where,
'X' is the random variable and |
|
k = Number of successes out of n
independent trials |
|
n
= Number of independent trials |
|
p
= Probability of success in a single trial |
|
q
= Probability of failure in a single trial |
|
p
+ q = 1 |
Important properties of Binomial distribution
|
1 |
Binomial
distribution is a discrete probability distribution, where the random
variable assumes a finite number of values 0, 1, 2... n. The distribution is specified
by two parameters ‘n’ and ‘p’. |
|
2 |
Mean
= np, variance (σ2) = npq |
|
3 |
Standard
deviation (σ) = (npq)(1/2) |
|
4 |
Skewness
(γ1) = (q – p) ÷ (npq)(1/2) |
|
5 |
Kurtosis
(γ2) = (1 – 6pq) ÷ (npq) |
|
6 |
If
‘x’ follows binomial distribution with parameters (n1,p) and ‘y’
follows binomial distribution with parameters (n2,p), and ‘x’ and
‘y’ are statistically independent, then (x + y) also follows binomial
distribution with parameters [(n1 + n2), p]. The result
can be extended to several independent binomial variates with a common ‘p’. |
|
Probability
function of Poisson distribution is |
|
P(X = k) = [e (−m).mk]
÷ k! |
|
Where,
'X' is the random variable and |
|
k = Number of successes out of n
independent trials |
|
n
= Number of independent trials (very large) |
|
p
= Probability of success in a single trial (very small) |
|
m = Mean, i.e. average rate of
success = np |
|
e = A mathematical constant =
2.718 (approx) |
Important properties of Poisson distribution
|
1 |
Poisson
distribution is a discrete probability distribution, where the random
variable assumes a countably infinite number of values 0, 1, 2... ∞. The
distribution is completely specified, when the parameter ‘m (positive)’ is
known. |
|
2 |
Mean
= m, variance (σ2) = m |
|
3 |
Standard
deviation (σ) = (m)(1/2) |
|
4 |
Skewness
(γ1) = 1 ÷ (m)(1/2) |
|
5 |
Kurtosis
(γ2) = 1 ÷ (m) |
|
6 |
If
‘x’ and ‘y’ are independent Poisson variables with parameters m1
and m2 respectively then (x + y) also follows Poisson distribution
with parameter (m1 + m2). |
|
Probability
function of Hypergeometric distribution is |
|
P(X
= k) = [ACk × (N – A)C(n – k)] ÷
[NCn] |
|
Where,
'X' is the random variable and |
|
N
= The fixed total population size |
|
A = The total number of
"successes" available in that population |
|
n = The number of draws (the sample size) |
|
k = The specific number of
“successes” in the sample out of total A “successes” in the population |
Example:
An urn
contains total 10 balls consisting of 4 red balls and 6 blue balls. If 3 balls
are drawn from the urn randomly without replacement, what is the probability
that exactly 2 of them are red?
Solution:
Here, N = 10,
A = 4, n = 3, and k = 2
Therefore,
P(X = 2) = [4C2 × (10 – 4)
C (3 – 2)] ÷ [10C3]
= (6 × 6) ÷ 120 = 36 ÷ 120 = 0.3 OR 30%
So
there is a 30%
probability of drawing
exactly 2 red balls.
Important properties of Hypergeometric distribution
|
1 |
Hypergeometric
distribution is a discrete probability distribution, where the random
variable assumes a finite number of values 0, 1, 2... m. |
|
2 |
It
has three parameters: N, n, and A. |
|
3 |
Mean
= n(A/N) |
|
4 |
Variance
= n(A/N)(1 – A/N){(N – n)/(N – 1)} |
|
5 |
|
If a random variable 'X' is normally
distributed with mean 'μ' and standard deviation 'σ', then Z = (X − μ)/σ is
called the "Standard Normal Variable".
In that case, P(X ≤ k) = Φ (Z) = Area
under "Standard Normal Curve" to the left of the ordinate at Z.
[Where, Z = (k − μ)/σ]
The probability curve of normal
distribution is known as Normal Curve. The curve is symmetrical and bell-shaped
and the two tails extend infinitely on either side.
Important properties of Normal distribution
|
1 |
Normal
distribution is a continuous probability distribution. |
|
2 |
It
has two parameters: μ and σ. |
|
3 |
Mean
= μ; Standard Deviation = σ |
|
4 |
Mean,
Median and Mode are equal, each being μ. In other words, Mean = Median = Mode
= μ |
|
5 |
The
quartiles are equidistant from mean. |
|
6 |
Approximately,
Q1 = μ – 0.67σ and Q2 = μ + 0.67σ |
|
7 |
Quartile
deviation = 0.67σ |
|
8 |
Mean
deviation = 0.80σ |
|
9 |
Skewness
(γ1) = 0; Kurtosis (γ2) = 0 |
|
10 |
The
normal curve is bell-shaped and symmetrical about the line x = μ. The two
tails of the curve extend to infinity on both sides of the mean. The maximum
ordinate is at x = μ and given by y = 1 ÷ [σ(2x)(1/2)] |
|
11 |
The
points of inflection of the normal curve are at x = μ ± σ. This means that at
these points the normal curve changes its curvature from concave to convex
and vice versa. |
|
12 |
Area
between Z = ± 1 is 68.27% Area
between Z = ± 2 is 95.45% Area
between Z = ± 3 is 99.73% |
|
13 |
If
x and y are independent normal variates with means μ1 and μ2,
and variances (σ1)2 and (σ2)2
respectively, then (x + y) is also a normal variate with mean (μ1
+ μ2) and variance [(σ1)2 + (σ2)2]. |
Probability Distributions
Selected Problems
Four coins are
tossed simultaneously. What is the probability of getting 2 heads and 2 tails?
Problem: 2
Find the
probability that in a family of 5 children there will be
(i)
At
least one boy,
(ii)
At
least one boy and one girl.
(Assume that
the probability of a female birth is ½).
Problem: 3
If a sample of
5 items is drawn randomly from a lot containing 10% defective items, what is
the probability of getting not more than one defective item?
Problem: 4
If 3% of the bolts manufactured by a company are defective, what is the
probability that in a sample of 200 bolts 5 will be defective? [Given, e−6
= 0.00248]
The average number of misprints per page of a book is
2. Assuming Poisson distribution, what is the probability that a particular
page is free from misprints? If the book contains 1000 pages, how many of the
pages contain more than 2 misprints?
Problem: 6
Suppose that the number of telephone calls an operator
receives from 11.00 a.m. to 11.05 a.m. follows a Poisson distribution with m =
3,
(i)
Find
the probability that the operator will receive no calls in that time interval
tomorrow.
(ii)
Find
the probability that in the next 3 days the operator will receive a total of 1
call in that time interval.
[Given
e = 2.718]
Problem: 7
The average number of defects per yard on a piece of
cloth is 0.9. What is the probability that a one-yard piece chosen at random
contains less than 2 defects? [Given e0.9 = 2.46]
Problem: 8
A system contains 1000 components. Each component
fails independently of the others and the probability of its failure in one
month is 1/1000. What is the probability that the system will function (i.e. no
component fails) at the end of the month?
Problem: 9
If 5% of the electrical bulbs manufactured by a
company are defective, use Poisson distribution to find the probability that in
a sample of 100 bulbs
(i)
None
is defective,
(ii)
5
bulbs will be defective
[Given
e−5 = 0.007]
Problem: 10
Find the probability that at most 5 defective bolts
will be found in a box of 200 bolts, if it is known that 2% of such bolts are
expected to be defective assuming Poisson distribution.
[Given e−4 = 0.0183]
Problem: 11
The manufacturer of a certain electronic component knows that 3% of his product is defective. He sells the components in boxes of 100 and guarantees that not more than 3 in any box will be defective. What is the probability that a box will fail to meet the guarantee? [Given e3 = 20.1]
Problem: 12
In a certain factory, blades are manufactured in packets
of 10. There is a 0.2% probability for any blade to be defective. Using Poisson
distribution, calculate approximately the number of packets containing 2
defective blades in a consignment of 20,000 packets. [Given e−0.02 =
0.9802]
Problem: 13
What is the probability of getting 3 white balls in a
draw of 5 balls from a box containing 5 white and 4 black balls?
Problem: 14
10 cards are drawn at random one by one without
replacement from a full pack of 52 playing cards. Find the mean and variance of
the number of red cards obtained.
The heights of a group of 2989 individuals are known
to be normally distributed with mean 65” and standard deviation 2.1”. Find the
number of individuals whose heights lie between 60.8” and 67.1”. Find also the number
of individuals whose heights are above 67.1”.
Problem: 16
The number of deaths per day in a city due to road
accidents and due to other causes independently follows Poisson distribution
with parameters 2 and 6 respectively. Find the probability that the total
number of deaths on a particular day is 2 or fewer. [Given e−4 =
0.018]
Problem: 17
A large
industrial firm allows a discount on any invoice that is paid within 30 days.
Of all invoices, 12% receive the discount. In a company audit, 8 invoices are
sampled at random. Find the probability that fewer than 2 receive the discount.
Problem: 18
Find an
approximate probability of getting more than 2 sickle cell anaemia patients in
a random sample of 200 individuals selected from a population having 0.5%
incidence of the disease.
Problem: 19
The
probability of a project manager meeting the deadline of completing a project
is estimated to be 0.8. If he has seven projects in hand, find the probability
that he can complete at most six projects by the deadline.
Problem: 20
A company
installs new central heating furnaces and has found that for 15% of all the
installations a return visit is needed to make some modifications. Six
installations were made in a particular week. What is the probability that, a
return visit will be needed in
(i)
All
of these cases?
(ii)
At
least one of these cases?
Problem: 21
70% of all the trucks undergoing a brake inspection at
a certain inspection facility pass the inspection. What is the probability that
out of 5 randomly selected trucks at least one will fail the inspection? How
many would you expect to fail the inspection?
Problem: 22
For a renowned airline company the record of flight
delay is very less: 1.2 cases a month, on an average. Find the probability of
more than 2 cases of flight delay in one month.
Problem: 23
Records indicate that, on an average, 3 breakdowns per
week occur on an urban highway during the morning rush hour on weekdays
consisting of 5 weekdays a week. Find the probability that on any given weekday
there will be fewer than two breakdowns on this highway during the morning rush
hour.
Problem: 24
The average number of traffic accidents on a certain
section of highway is 2 per week. Find the probability of no accidents on this
section of highway during a one-week period. What would be the variance of
number of traffic accidents per week?
Problem: 25
A container packaging machine fills containers such
that weights are normally distributed with a mean of 1.2 lb and an SD of 0.2
lb. The container level states the weight as 1 lb. Obtain the probability that
a randomly selected container contains less than the promised weight. [Consider
φ (1) = 0.84]
Problem: 26
The minimum daily temperature in a certain city
follows normal distribution with a mean of 15OC and an SD of 3OC
in winter season. What is the probability that, the minimum temperature on a
randomly selected winter day will be
(i)
More
than 21OC?
(ii)
Between
12OC and 18OC?
[Given
that φ (1) = 0.8413 and φ (2) = 0.9772]
Problem: 27
A person is considered hypothalamic if his or her potassium level in blood is as low as 3.5 mEq/Ls (milliequivalents per liter). An individual's daily potassium level in the blood is assumed to follow a normal distribution with a mean of 3.8 mEq/Ls and a standard deviation of 0.2 mEq/Ls. What is the probability that in a randomly selected day his potassium level will suggest that he is hypothalamic? [Given that φ (1.5) = 0.9332]
Assuming that the rate of normal heart beat in healthy
individuals is normally distributed with mean of 70 beat/minutes and a standard
deviation of 10 beat/minutes. If an individual is selected at random, what is
the probability that his heart beat is
i)
Above
80 beats/minute;
ii)
Between
50 and 90 beats/minute
[Given, φ (1) = 0.8413 and φ (2) = 0.9772]
