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DESCRIPTION OF THE TOPIC
INTRODUCTION:
We live in a world in which we are unable to forecast the future with complete certainty. Our
need to cope with uncertainty leads us to the study and use of probability theory. By organizing
the information and considering it systematically, we will be able to recognize our assumptions,
communicate our reasoning to others, and make a better decision than we could by using a
shot-in-the-dark approach. Probability is a part of our everyday lives and in personal decisions
also we face uncertainty and use probability theory whether or not we admit the use of
something advanced. When we hear a weather forecast of a 60 percent chance of rain, we
change our plans from outdoor game to indoor game. The managers who deal with inventories
of highly styled women’s clothing must wonder about the chances that sales will reach or
exceed a certain level. Probability deals with many uncertainties in research and probability
theory is the foundation for statistical inference.
LEARNING OBJECTIVES:
The learners are expected to know the following from this module:
o Fundamentals of probability
o Basic concept of random variables and probability distributions
o Various probability distributions and their applications used in social science research
WHAT IS PROBABILITY?
The probability of an event has been defined as its long-run relative frequency. It has also been
thought of as a personal degree of belief that a particular event will occur (subjective
probability). A probability provides a quantitative description of the likely occurrence of a
particular event. Probability is conventionally expressed on a scale from 0 to 1; an impossible
event has a probability of 0, and a definite event has a probability of 1.
Items Description of Topic
Course Data Analysis for Social Science Teachers
Topic Different Types of Distributions
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In some experiments, all outcomes are equally likely. For example, if you are throwing a six
faced dice, all six faces are equally likely to appear, which means all of them have the same
probability to appear. All these six events are called equiprobable events.
𝑃(𝐸) =
𝑁𝑢𝑚𝑏𝑒𝑟𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Mutually Exclusive Events:
Two events are mutually exclusive if it is impossible for them to occur together. Two events A
and B can be mutually exclusive if and only if P (𝐴 ∩ 𝐵) = 0.
Additive Law of Probability:
The addition rule is a result used to determine the probability that event A or event B occurs,
or both occur. The result is often written as follows, using set notation:
)()()()( BAPBPAPBAP .
Conditional Probability:
In many situations, once more information becomes available, we are able to revise our
estimates for the probability of further outcomes or events happening. For example, suppose
you go out for lunch at the same place and time every Friday and you are served lunch within
15 minutes with probability 0.9. However, given that you notice that the restaurant is
exceptionally busy, the probability of being served lunch within 15 minutes may reduce to 0.7.
This is the conditional probability of being served lunch within 15 minutes given that the
restaurant is exceptionally busy.
The usual notation for "event A occurs given that event B has occurred" is "A / B" (A given
B). The symbol | is a vertical line and does not imply division. P(A / B) denotes the probability
that event A will occur given that event B has occurred already.
A rule that can be used to determine a conditional probability from unconditional probabilities
is:
)(
)(
)/(
BP
BAP
BAP
.
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Multiplicative Law of Probability:
The multiplicative law is a result used to determine the probability that two events, A and B,
both occur. The multiplicative law follows from the definition of conditional probability. The
result is often written as follows, using set notation:
)/().()()./()( ABPAPBPBAPBAP .
Independent Events:
Two events are independent if the occurrence of one of the events gives us no information
about whether or not the other event will occur; that is, the events have no influence on each
other. It is denoted by:
)().()( BPAPBAP .
RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS:
Random variables:
The outcome of an experiment need not be a number, for example, when a coin is tossed the
outcome can be 'head' or 'tail'. However, we prefer to represent the outcomes as numbers. A
random variable is a function that assigns a unique numerical value with every outcome of an
experiment. The value of the random variable will vary from trial to trial as the experiment is
repeated.
There are two types of random variables -- Discrete and Continuous. A random variable has
either an associated probability distribution (Discrete random variable) or probability density
function (Continuous random variable).
Expected Value for a Discrete Distribution:
Expected value or mean for a discrete distribution is computed by using the following formula.
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)()( xxPxE
where:
E(x) = Expected value of the random variable
x = Values of the random variable
P(x) = Probability of the random variable taking on the value of x
Variance:
The (population) variance of a random variable is a non-negative number which gives an idea
of how widely spread are the values of the random variable; the larger the variance, the more
scattered the observations on average. Stating the variance gives an impression of how closely
concentrated is the distribution round the expected value; it is a measure of the 'spread' of a
distribution about its average value.
Variance is symbolized by V(X) or Var(X) or 2
and it is computed by using the formula
given below.
)()}({ xPxEx 2
x
where:
E(x) = Expected value of the random variable
X = Values of the random variable
P(x) = Probability of the random variable having the value of x
PROBABILITY DISTRIBUTIONS:
The following are popular probability distributions.
1. Binomial Distribution
2. Poisson Distribution
3. Normal Distribution
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Binomial and Poisson distributions are classified as Discrete probability distribution, whereas
normal distribution is a continuous probability distribution.
BINOMIAL DISTRIBUTION
Binomial Distribution is one of the theoretical or expected frequency distributions in
probability distributions. It is also known as “Bernoulli distribution”, since it is associated with
the name of Swiss Mathematician “James Bernoulli”, who is also known as Jacques or Jacob.
The Binomial distribution expresses the probability of one set of dichotomous alternatives, i.e.,
success or failure.
In Bernoulli process, an experiment is performed repeatedly, yielding either a success or a
failure in each trial and where there is absolutely no pattern in the occurrence of success and
failures. The trials are independent in this process and the mathematical model for this process
is developed under a very specific set of assumptions involving the concept of a series of
experimental trials. These assumptions are:
(i) An experiment, under the same conditions, is performed for ‘n’ number of trials.
(Where ‘n’ is finite)
(ii) There are only two possible outcomes of the experiment in each trial, and the sample
space for each experiment is denoted by ‘s’ where S= {success, failure}.
(iii) The probability of success is denoted by ‘p’, which remains constant from trial to
trial and the probability of failure is denoted by q=(1-p).
(iv) The trials, which are statistically independent, do not affect the outcomes of
subsequent trials.
From the following example, we can see how the binomial distribution arises.
If a coin is tossed once, there are two outcomes, tail or head. The probability of obtaining a
head, p=1/2 and the probability of obtaining a tail, q=1/2. These are terms of binomial (q+p),
where (q+p)=1.
In general, in ‘n’ tosses of a coin, the probabilities of the various possible events are given by
the successive terms of the Binomial expansion. The general form of the Binomial distribution
is denoted as follows.
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xnx
qp
xnx
n
xP
)!(!
!
)(
n = sample size
x = Number of successes
n - x = Number of failures
p = Probability of a success
q = 1 - p = probability of a failure
n! =n(n - 1)(n - 2) . . . (2)(1)
x! = x(x - 1)(x - 2) . . . (2)(1)
0! = 1
Mean of the binomial distribution=np
Variance of the distribution = npq
POISSON DISTRIBUTION:
Poisson distribution, which is also a discrete probability distribution, was developed by a
French mathematician SD Poisson (1781-1840) and hence named after him. Along with the
Normal and Binomial distributions, the Poisson distribution is one of the most widely used
distributions. It is used in quality control statistics to count the number of defective items or
in insurance problems to count the number of casualties or in waiting-time problems to count
the number of incoming telephone calls or incoming customers or the number of patients
arriving to consult a doctor in a given time period, and so forth. All these examples have a
common feature: they can be described by a discrete random variable, which takes on integer
values (0, 1, 2, 3 and so on).
The characteristics of the Poisson distribution are:
1. The events occur independently. This means that the occurrence of a subsequent even
is not at all influenced by the occurrence of an earlier event.
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2. Theoretically, there is no upper limit with the number of occurrences of an event during
a specified time period.
3. The probability of a single occurrence of an event within a specified time period is
proportional to the length of the time period of interval.
4. In an extremely small portion of the time period, the probability of two or more
occurrences of an event is negligible.
The Poisson distribution is used for modeling rates of occurrence. The Poisson distribution
has one parameter:
!
)(
)(
x
e
xP
x
Where:
x = Number of successes
= Expected number of successes
e =Base of the natural number system (2.71828)
For Poisson distribution, Mean = Variance= λ
NORMAL DISTRIBUTION:
The preceding two distributions discussed in this chapter are discrete probability distributions.
We shall now take up another distribution in which the random variable can take on any value
within a given range. This is the normal distribution, which is an important continuous
probability distribution. This distribution is also known as the Gaussian distribution after the
name of the eighteenth century mathematician-astronomer Karl Gauss, whose contribution in
the development of the normal distribution was very considerable. As a vast number of
phenomena have approximately normal distribution, it has to make inferences by drawing
samples. The normal distribution has certain characteristics, which make it applicable to such
situations.
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22
2/)(
2
1
)(
x
exf
Where:
x = Any value of the continuous random variable (minus infinity to plus infinity)
= Population standard deviation
e = Base of the natural log = 2.7183
= Population mean
Probability for different X values can be obtained from normal distribution table available at
the end of all the units.
CHARACTERISTICS OF NORMAL PROBABILITY DISTRIBUTION:
Normal Probability Distribution
Let us see what this figure indicates in terms of characteristics of the normal distribution. It
indicates the following characteristics.
1. The curve is bell shaped, that is, it has the same shape on either side of the vertical line
from mean.
2. It has a single peak. As such it is uni-model.
3. The mean is located at the centre of the distribution.
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4. The distribution is symmetrical
5. The two tails of the distribution extend indefinitely but never touch the horizontal axis.
6. Since the normal cure is symmetrical, the lower and upper quartiles are equidistant from
the median, that is, Q3-Median=Median-Q1.
7. The mean, median, and mode have the same value, that is, mean=median-mode
Area under the normal curve:
Area between μ+1σ to μ-1σ is 68.26%
Area between μ+2σ to μ-2σ is 95.44%
Area between μ+3σ to μ-3σ is 99.72%
The most important reasons for its applicability:
1) Normal distribution is important because a wide variety of naturally occurring random
variables such as heights and weights of all creatures are distributed evenly around a
central value, average, or norm (hence the name normal distribution). Although the
distributions are only approximately normal, they are quite close. Whenever too many
factors influence the outcome of a random variable, the underlying distribution is
normal.
Ex: Height of a tree is determined by the “sum” of such factors as rain, soil quality,
sunshine, disease, etc.
μ±1σ
μ±2σ
μ±3σ
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2) All the statistical tables are limited by the size of their parameters (Mostly). However,
when these parameters are large enough one may use normal distribution for calculating
the critical values-for these tables.
3) Approximation to Binomial is made by taking npqnp 2
; .
Application: - The probability of a defective item coming off a certain assembly line is
p=0.25. A sample of 400 items is selected from a large lot of these items. What is the
probability of getting 90 (or) less items are defective?
4) If the mean and S.D of a normal distribution are known, it is easy to convert back and
forth from the raw scores to percentiles.
5) It has been proven that the underlying distribution is normal if and only if the sample
mean is independent of sample variance. This characterizes the normal distribution.
Therefore, many effective transformations can be applied to convert almost only shaped
distribution into a normal one.
6) The most important reason for popularity of normal distribution is the Central Limit
Theorem (CLT). The distribution of the sample averages of a large number of
independent variables (random) will be approximately normal regardless of the
distributions of the individual random variables. The CLT is useful especially when
you are dealing with a population with an unknown distribution.
7) The normality condition is required by almost all kinds of parametric statistical tests.
Using most statistical tables, such as T-table, and F-table, all require the normality
condition of the population.
8) This condition should be tested before using the tables, otherwise the conclusion will
be wrong.
A normal curve with mean µ and standard deviation σ can be converted into a standard
normal distribution by performing the change of the scale and origin. The original µ and σ
will be converted to 0 and 1 respectively. The units for the standard normal distribution curve
are denoted by Z and are called the Z values or Z scores. They are also called standard units
or standard scores. The Z score is known as a ‘standardized’ variable because it has a zero
mean and a standard deviation of one.
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x
z
This transformation will be used widely in Z test in hypothesis testing.
where:
X = Any point on the horizontal axis
= Standard deviation of the normal distribution
= Population mean
Z = Scaled value (the number of standard deviations a point x is from the mean)
REFERENCES:
1. Levin, I. Richard and Rubin S. David. “Statistics for management”, P H I, New Delhi,
2000.
2. Sancheti, D.C., and Kapoor, V.K. , “Business Statistics”, New Delhi
3. Jaggia, S., & Kelly, A. (2013). Business statistics: Communicating with numbers.
McGraw-Hill Education.
4. Gupta, S.P., (2012). Statistical Methods. Sultan Chand & Sons.
5. Spiegel, M. R., Schiller, J., & Srinivasan, R. A. (2017). Probability and Statistics.
Schaum’s outline series in Mathematics. McGraw-Hill Education.
6. Ross, S. (2019). A first course in probability. Pearson.
For more information on the above topic, please go through the following OERs.
https://courses.lumenlearning.com/introstats1/chapter/the-terminology-of-probability/
https://courses.lumenlearning.com/sanjacinto-finitemath1/chapter/5-9-probability-
distributions/
