Probability, it is discussed for business statistics

1. Probability
Dr. R. Muthukrishnaveni
Assistant Professor

2. Introduction
• Probability has its origin in the games of chance
related to gambling such as tossing of a coin,
throwing of a die, drawing cards from a pack of
cards etc
• Jerame Cardon, an Italian mathematician wrote
‘ A book on games of chance’ which was
published on 1663.
• Starting with games of chance, probability has
become one of the basic tools of statistics.

3. Introduction
• The knowledge of probability theory makes it
possible to interpret statistical results, since
many statistical procedures involve conclusions
based on samples.
• Probability theory is being applied in the
solution of social, economic, business problems.
• the mathematical theory of probability has
become the basis for statistical applications in
both social and decision-making research.

4. Basic concepts
• Random experiment
• Trial
• Outcomes
• Event
• Sample space
• Equally likely events
• Mutually exclusive events
• Exhaustive events
• Complementary events
• Compound events
 Independent event
 Dependent event

5. Types of Probability
• Mathematical Probability or a priori
probability or a classical probability
 P(A) = m/n
• Statistical Probability or a posteriori
probability or a empirical probability.
 P(A) = Limit (m/n) n => ∞

6. Probability Theorem
• Addition theorem
▫ Mutually exclusive P(A or B) = P(A) + P(B)
▫ Not Mutually exclusive P(A or B) = P(A) + P(B) –
P(A∩B)
▫ Conditional Probability P(B/A) = P(A∩B)/P(A)
• Multiplication theorem
▫ Independent events P(A∩B) = P(A) x P(B)
▫ Dependent events P(A∩B) = P(A) x P(B/A)

7. BAYES’ Theorem

8. Permutation
• nPr =n!/(n-r)!
• the number of arrangements that can be made out of n
things taken r at a time is known as the number of
permutation of n things taken r at a time and is
denoted as nPr.
A,B,C arranged in AB, BA, BC, CB, AC, CA
3P2 = 3x2x1/1 =6

9. Combinations
• nCr = nPr/r! Or n!/ (n-r)!r!
• Out of three things A,B,C we have to select two
things at a time. This can be selected in three
different ways as follows:
A B, AC, B C
3C2 = 6/2 x 1 = 3

10. Variable
It has been a general notion that if an
experiment is conducted under identical
conditions, values so obtained would be similar.
Observations are always taken about a factor or
character under study, which can take different
values and the factor or character is termed as
variable.

11. Random Variable
These observations vary even though the
experiment is conducted under identical
conditions. Hence, we have a set of outcomes
(sample points) of a random experiment. A rule
that assigns a real number to each outcome
(sample point) is calledrandom variable.

12. Discrete Random Variable
If a random variable takes only a finite or a
countable number of values, it is called a
discrete random variable.
For example, when 3 coins are tossed, the
number of heads obtained is the random
variable X assumes the values 0,1,2,3 which
form a countable set. Such a variable is a
discrete random variable.

13. Continuous Random Variable
A random variable X which can take any
value between certain interval is called a
continuous random variable.
For example the height of students in a
particular class lies between 4 feet to 6 feet.

14. Probability Distribution
From the above discussion, it is clear that
there is a value for each outcome, which it takes
with certain probability. Hence a list of values of
a random variable together with their
corresponding probabilities of occurrence, is
termed as Probability distribution.
As a tradition, probability distribution is
used to denote the probability mass or
probability density, of either a discrete or a
continuous variable.

15. Probability Distribution
 Theoretical listing of outcomes and probabilities
(mathematical models)
 Empirical listing of outcomes and their observed
frequencies
 A subjective listing of outcomes associated with
their subjective or contrived probabilities
representing the degree of conviction of decision
maker

16. THEORETICAL DISTRIBUTIONS
• 1st Kind of probability distribution
• Some important theoretical distribution
▫ Binomial Distribution
▫ Multinomial Distribution
▫ Negative Binomial Distribution
▫ Poisson Distribution
▫ Hyper Geometric Distribution
▫ Normal Distribution

17. Binomial Distribution
• Swiss mathematician James Bernoulli also
known as Jaccques or Jakob (1654 – 1705)
• P(r) = nCr qn-r pr
• Mean = np
• S.D. = √npq

18. Characteristics of Binomial
Distribution
• Binomial distribution is a discrete distribution in which the
random variable X (the number of success) assumes the values
0,1, 2, ….n, where n is finite.
• Mean = np, variance = npq and standard deviation s =√ npq ,
• The mode of the binomial distribution is that value of the
variable which occurs with the largest probability. It may have
either one or two modes.
• If two independent random variables X and Y follow binomial
distribution with parameter (n1, p) and (n2, p) respectively, then
their sum (X+Y) also follows Binomial
• distribution with parameter (n1 + n2, p)
• If n independent trials are repeated N times, N sets of n trials are
obtained and the expected frequency of r success is N(nCr pr qn-
r). The expected frequencies of 0,1,2… n success are the
successive terms of the binomial distribution of N(q + p)n

19. Multinomial Distribution
n!
P(r1 ,r2, ..rk ) = ________ pr1pr2..prk
(r1!r2!rk!)

20. Negative Binomial Distribution
-r
f(x;r,p) = x pr (-q)x
X = 0,1,2....

21. Poisson Distribution
e-m mr
• P(r) = r!
• r = 0,1,2,3...

22. Characteristics of Poisson Distribution:
• Discrete distribution: Poisson distribution is a discrete distribution
like Binomial distribution, where the random variable assume as a
countably infinite number of values 0,1,2 ….
• The values of p and q: It is applied in situation where the probability
of success p of an event is very small and that of failure q is very
high almost equal to 1 and n is very large.
• The parameter: The parameter of the Poisson distribution is m. If
the value of m is known, all the probabilities of the Poisson
distribution can be ascertained.
• Values of Constant: Mean = m = variance; so that standard
deviation = m Poisson distribution may have either one or two
modes.
• Additive Property: If X and Y are two independent Poisson
distribution with parameter m1 and m2 respectively. Then (X+Y)
also follows the Poisson distribution with parameter (m1 + m2)

23. Characteristics of Poisson Distribution:
• As an approximation to binomial distribution: Poisson
distribution can be taken as a limiting form of Binomial
distribution when n is large and p is very small in such a
• way that product np = m remains constant.
• Assumptions: The Poisson distribution is based on the
following assumptions.
• i) The occurrence or non- occurrence of an event
does not influence the occurrence or non-occurrence of any
other event.
• ii) The probability of success for a short time
interval or a small region of space is proportional to the length
of the time interval or space as the case may be.
• iii) The probability of the happening of more than
one event is a very small interval is negligible.

24. Normal distribution

25. Condition of Normal Distribution
• Normal distribution is a limiting form of the
binomial distribution under the following
conditions.
a) n, the number of trials is indefinitely large ie.,
nà ¥ and
b) Neither p nor q is very small.
• Normal distribution can also be obtained as a
limiting form of Poisson distribution with parameter
mà ¥
• Constants of normal distribution are mean=m,
variation=s2, Standard deviation=s

26. Normal probability curve

27. Properties of normal distribution
• The normal curve is bell shaped and is
symmetric at x = m.
• Mean, median, and mode of the distribution are
coincide i.e., Mean = Median = Mode = m
• It has only one mode at x = m (i.e., unimodal)
• Since the curve is symmetrical, Skewness = b1 =
0 and Kurtosis = b2 = 3.
• The points of inflection are at x = m ± s
• The maximum ordinate occurs at x = m and its
value is = 1/s 2p

28. Properties of normal distribution
• The x axis is an asymptote to the curve (i.e. the curve
continues to approach but never touches the x axis)
• The first and third quartiles are equidistant from
median.
• The mean deviation about mean is 0.8 s
• Quartile deviation = 0.6745 s
• If X and Y are independent normal variates with mean
m1 and m2, and variance s1
2 s22 respectively then their
sum (X + Y) is also a normal variate with mean (m1 +
m2) and variance (s12 + s22 )
• Area Property P(m - s < ´ < m + s) = 0.6826 P(m - 2s < ´
< m + 2s) = 0.9544
• P(m - 3s < ´ < m + 3s) = 0.9973