PROBABILITY: • “Probability measures provide the decision maker in business and in government with the means for qualifying the uncertainties which affect his choice of appropriate actions.” -Morris Hamburg • One of the primary reasons for the development of the theory of probability is the presence in almost every aspect of life, of random phenomenon. A phenomenon is random if chance factors determine its outcomes. All the possible outcomes may be known in advance, but the particular outcome of a single trial in any experimental operation cannot be pre -determined. Nevertheless, some regularity is built into the process so that each of the possible outcomes can be assigned a probability fraction. • Probability is especially important in statistics because of the many principles and procedures that are based on this concept. Reasoning in terms of probabilities is one weapon by which we attempt to reduce this uncertainty or ignorance. In statistics, probability is a numerical value that measures the uncertainty that a particular event will occur. THEORIES OF PROBABILITY: • The solution to many problems involving probabilities requires a thorough understanding of some of basic rules which govern the manipulation of probabilities. They are generally called probability theorems.

1. UNIT 4
STATISTICS:
(Theory Of Probability- Principles And Properties Of Normal Distribution-
Binomial Distribution- Interpretation Of Data Using The Normal Probability
Curve- Causes Of Distribution- Deviations From The Normal Forms)
SUBMITTED TO: SUBMITTED BY:
DR. SATISK K. HIMANI BANSAL
MVSCOSH MASLP 1ST YEAR

2. “
”
“Probability measures provide the decision maker in business and in
government with the means for qualifying the uncertainties which
affect his choice of appropriate actions.”
-Morris Hamburg
PROBABILITY

3. THEORIES OF PROBABILITY
Addition
Theorem
• If two events are mutually exclusive and the probability of the
one is P1, while that of the other is P2, the probability of either
the one or the other occurring is the sum P1+P2
Multiplication
Theorem
• If two events are mutually independent and the probability of
the one is P1, while that of the other is P2, the probability of
the two events occurring simultaneously is the product of P1
and P2
Theorem of
Conditional
Probability
• The probability that both of two dependent sub-events can
occur is the product of the probability of the first sub-event
and the probability of the second after the first sub-event has
occurred
Bayes’
Theorem
• The concern for revising probabilities arises from a need to
make better use of experimental information. This is referred
to as Bayes’ Theorem

5. PROBABILITY DISTRIBUTION
Normal
Distribution
Chi Square
Distribution
Binomial
Distribution
Poisson
Distribution
A list describing all the possible outcomes
of a random variable along with their
corresponding probabilities of occurrence
In simple terms, it shows the likelihood of
different possible outcomes of a variable

8. STANDARD NORMAL DISTRIBUTION & STANDARD
SCORES
Z- distribution/Standard normal distribution: a
special case of the normal distribution where
the mean is zero and the standard deviation is 1
A value on the standard normal distribution is
known as a standard score or a Z-score
Z- score: represents the number of standard
deviations above or below the mean that a
specific observation falls

9. STANDARDIZATION: HOW TO CALCULATE Z- SCORE
Standardization: to take observations drawn from normally
distributed populations that have different means and
standard deviations and place them on a standard scale.
To standardize our data, we need to convert the raw
measurements into Z-scores.
Here, X represents the raw value of the measurement of
interest. Mu and sigma represent the parameters for the
population from which the observation was drawn.

10. FINDING AREA UNDER THE NORMAL DISTRIBUTION
CURVE
The proportion of
the area that falls
under the curve
between two points
on a probability
distribution plot
indicates the
probability that a
value will fall within
that interval
We can calculate
areas by looking up
Z-scores in a
Standard Normal
Distribution Table
We can transform
the values from any
normal distribution
into Z-scores, and
then use a table of
standard scores to
calculate
probabilities

11. IMPORTANCE OF NORMAL DISTRIBUTION CURVE

12. CENTRAL
LIMIT
THEOREM
For a sufficiently large sample size, the sampling distribution
of the mean for a variable will approximate a normal
distribution regardless of that variable’s distribution in the
population
As the sample size increases, the sampling distribution
converges on a normal distribution where the mean equals
the population mean, and the standard deviation equals
σ/√n. Where: σ = the population standard deviation; n = the
sample size
In a population, the values of a variable can follow different
probability distributions. These distributions can range from
normal, left-skewed, right-skewed, and uniform among
others.

13. BINOMIAL DISTRIBUTION

14. BINOMIAL DISTRIBUTION

15. QUESTIONS ASKED IN PREVIOUS YEARS
1. The incidence of occupational disease in an industry is such that the workmen have a 20% chance of suffering from it. What is the probability
that out of six workmen 4 or more will contact the disease? (8 marks, 2019)
2. Write a short note on Binomial Distribution. (4 marks, 2018)
3. Narrate the properties and importance of Normal Distribution. Given the mean and SD of serum iron values for healthy men are 120 and 150
micrograms. In a sample of 500 cases, assuming normality, estimate the number of individuals with serum iron values less than 135
microgram. Also, find the probability that people will get serum value from 90 to 135 microgram? (16 marks, 2018)
4. Explain the properties and importance of the normal distribution. (8 marks, 2017)
5. A study on the FBS levels of 100 diabetics gave a mean of 156 mg/dl and a SD of 12 mg/dl. Assuming a normal distribution, what is the
probability that any given individual will have an FBS level between 132 and 168 mg/dl? Estimate the number of subjects with FBS levels less
than 168 mg/dl. (8 marks, 2017)
6. Explain the properties of Normal distribution and Binomial distribution. (12 marks, 2016)
7. Suppose the average length of stay in a chronic disease hospital of a certain type of the patient is 60 days with a standard deviation of 15. By
assuming a normal distribution, find the probability that a randomly selected patient will have a length of stay between 30 and 60 days. (4
marks, 2016)
8. What is the importance of normal distribution? State its properties. (8 marks, 2013)
9. Short note on normal probability distribution. (8 marks, 2012, 2011)

16. REFERENCES
1. Statistics, Gupta B.N. (1991)
2. https://www.britannica.com/science/probability-theory/An-alternative-interpretation-of-probability
3. https://www.cuemath.com/data/probability-theory/
4. https://egyankosh.ac.in/bitstream/123456789/23467/1/Unit-4.pdf
5. https://statisticsbyjim.com/basics/normal-distribution/
6. https://statisticsbyjim.com/basics/central-limit-theorem/
7. https://www.wallstreetmojo.com/law-of-large-numbers/