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Theory of Probability
The theory of probability provides a means of getting an idea of the likelihood of occurrence of different
events resulting from a random experiment in terms of quantitative measures ranging between 0 and 1.
The probability is zero for an impossible event and one for an event which is certain to occur. In other
words probability is a concept which numerically measures the degree of uncertainty and therefore of
certainty of the occurrence of events.
Terminologies used in Probability
Random Experiments: If in each trial of an experiment conducted under identical conditions the
outcome is not unique but may be any of the possible outcome then such an experiment is called a
random experiment. Example: If we toss a coin, only two possible outcomes either Head or Tail.
Deterministic experiment: The outcomes of such experiment are always certain.
Sample space: The set of all possible outcomes.
Sample point: Elements of sample space.
Trial: The performance of random experiment for one time is called a trial.
Event: The outcome of trial is called an event.
Equally likely event: Two or more events are said to be equally likely if the chances of their happening
are same.
Independent event: Two or more events are said to be independent event if the happening of one does
not affect the happening of others.
Dependent event: Two or more events are said to be dependent event if the happening of one affect the
happening of others.
Impossible event: An event which cannot be happen.
Mutually exclusive event: Two or more events are said to be mutually exclusive event if the happening
of one prevents the happening of others or A and B are mutually exclusive if A  B  
Exhaustive events: All possible outcomes of random experiments are called exhaustive events or A and
B are exhaustive events if A  B  S
Exhaustive and mutually exclusive event: Events E1 , E2 ,....E n are mutually exclusive and exhaustive
if E1  E2  ....  En  S and E i  E j    i  j
Equally likely outcomes: All outcomes with equal probability.
Favorable cases: The outcomes from all equally likely outcomes of random experiment, which are
favorable for the happening of a particular event, are called favorable cases.
Probability: Probability of happening of an event out of the events occurring in various equally likely
ways is the ratio of favorable ways of occurrence of the event to the total number of events.
No. of favourable cases
Total no. of equally likely cases
Conditional probability: When the happening of event A depends on the happening of another event B ,
the probability of event A is called conditional probability and is denoted by PA B 
Mathematically p 
PA B  
P A  B 
P B 
Laws of Probability
Additive Law:
 P A  B  P A  PB  P A  B
 P A  B  P A  PB if A and B are mutually exclusive
Multiplicative Law:
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 
 
 Dependent event: P A  B  P AP B A  PBP A B
 Independent event: P A  B  P APB
Question: Find the chance that if a card is drawn at random from an ordinary pack, it is one of the court
(Ans: 3
cards.
)
13
Question: An urn contains 9 balls, two of which are red, three blue and four black. Three balls are drawn
from the urn at random, that is every ball has an equal chance of being included in the three. What is the
chance that
(Ans: 2 )
a) Three balls are of different colours.
7
(Ans: 55
b) Two balls are of same colour and third of different.
(Ans: 5
c) The balls are of the same colour.
)
84
84
)
Question: A bag contains 4 red and 3 blue balls. Two drawing of two balls are made. Find the chance
that the first drawing gives 2 red balls and the second drawing 2 blue balls.
(Ans: 2
a) If the balls are returned to the bag after the first draw.
)
49
(Ans: 3 )
35
b) If the balls are not returned.
Question: A can solve 75% of the problems of this book and B can solve 70% . What is the probability
(Ans: 37
that either A or B can solve a problem selected at random.
40
)
Question: There are 6 red and 4 green balls in a bag. Two balls are drawn one after other without
(Ans: 1 )
replacement. What is the chance that both will be red?
3
Baye’s Theorem:
Let B1 , B2 ,....Bn be n mutually exclusive and exhaustive events whose union be sample space and
assume
A be an arbitrary event in space and P A  0 and PB1 , PB2 ,....PBn  and
P AB1 , PA B2 ,....PA Bn  are known, then
PBi A 
PBi P A Bi 
 PB PA B 
n
i 1
i
i
Question: A bag contains 2 white and 3 red balls and a bag B contains 4 white and 5 red balls. One
ball is drawn at random from one of the bag and is found to be red. Find the probability that it was drawn
(Ans: 25
from bag B.
52
)
Question: A factory has 3 machines A, B, C, producing 1000 , 2000 , 3000 bolts per day respectively.
A produces 1% defective, B 1.5% and C 2% defective. A bolt is checked at random at the end of a day
and is found to be defective. What is the probability that it came from machine A?
Question: The contents of three urns are as follows
White ball
Urn 1 1
Urn 2 2
Urn 3 3
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Red ball
2
3
1
(Ans: 1
10
)
Black ball
3
1
2
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An urn is selected At random and from it two balls are drawn at random. The two balls are one red and
(Ans: 6
one white. What is the probability that they come from second urn.
11
)
Binomial Distribution
If a trial is repeated n times under the same conditions, such that probability of success of an event ‘p’
remains constant in all these trials, then the probability of ' r ' success P X  r  ( r is integer) is given
by
P X  r nCr p r q n  r , r  0,1,2.....n
where q is the probability of failure and Binomial distribution is given by  p  q  where q  1  p
Characteristics of Binomial Distribution
 Mean of Binomial distribution is np .
 Variance of Binomial distribution is npq .
n
n  r  p Pr 
r  1q
2
2


q  p
1  2 p


 Recurrence relation for Binomial distribution is Pr  1 
 32
 Coefficient of skewness is 1  3
npq
npq
2

1  6 pq
 Coefficient of kurtosis is  2  42  3 
npq
2
Question: Six dice are thrown 729 times. How many times do you expect at least three dice to show a
five or six?
(Ans: 233 )
Question: A die is thrown 8 times and it is required to find the probability that 3 will be shown
a) Exactly two times.
b) Atleast seven times.
c) Atleast once.
6
(Ans: 28  5
(Ans: 41
68
)
68
)
 6 )
(Ans: 1 5
8
Question: A bag contains 10 balls each marked with one of the digits 0 to 9 . If four balls are drawn
successively with replacement from the bag. What is the probability that none is marked with digit 0 ?
 10 )
(Ans: 9
4
Question: A bag contains 5 white, 7 red and 8 black balls. If four balls are drawn one by one with
replacement. What is the probability that
a) None is white
(Ans: 81
b) All are white
(Ans: 1
c) Atleast one is white
d) Only two are white
256
)
)
256
(Ans: 175
)
256
(Ans: 27
)
128
Question: Fit a Binomial Distribution for the following data
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x
y
0
2
1
14
2
20
3
34
4
22
5
8
(Ans: 2,10,26,34,22,6 )
Poisson Distribution
The Poisson distribution is limiting case of Binomial distribution when p is very small and n is large
enough so that np is a finite constant say m . The probability of r success is given by
P X  r  
e m m r
,
r!
0r n
Characteristics of Poisson distribution
 Mean of Poisson distribution is m .
 Variance of Poisson distribution is m .
 Recurrence relation for Poisson distribution is Pr  1 
m
Pr 
r  1
Question: Six coins are tossed 6400 times. Find the approximate probability of getting six heads r times
and 2 times.
(Ans:
100r e 100 , 1002 e 100 )
r!
2!
Question: If the probability that an individual suffers a bad reaction from a certain injection is 0.001 ,
determine the probability that out of 2000 individuals
4e 2
a) Exactly three.
(Ans:
)
3
b) More than two.
(Ans: 1  5e 2 )
c) None.
(Ans: e 2 )
d) Atleast one.
(Ans: 1  e 2 )
individuals will suffer bad reaction from that injection.
Question: If 3% of electric bulbs are manufactured by a company is defective. Find the probability that
in a sample of 100 bulbs exactly 5 bulbs are defective.
Question: Fit a Poisson Distribution for the following data
x
0 1
2
3
y
21 18 7
3
(Ans:
81 3
e )
40
4
1
(Ans: 20,18,8,2,1 )
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Normal Distribution
It is a continuous probability distribution. It is the limiting case of Binomial distribution for large values
of n and neither the probability of success or failure is very small. The probability density function of
Normal distribution is given by
f x  
1
 2
e
1  x 
 

2  
2
where the variable x is called normal variate and
can assume all values from   to   ,  and
 are called the parameters of distribution are
respectively the mean and standard deviation of
the distribution
The normal distribution graph is called normal
curve, which is bell shaped and symmetrical
about the mean. The line x  mean divides the
area under the normal curve about X-axis into
two equal parts. Thus the median coincides with
mean and mode. The total area under normal
curve above X-axis is one.
Characteristics of Poisson distribution
 Area under the normal curve is one.
 Mean, Median and Mode coincides at the origin.
 The odd moments about mean are are zero.  2 n 1  0
 The even moments about mean are given by  2 n   2 2n  1 2 n 2
or  2 n   2 n 2n  12n  3.........3.1
 Coefficient of skewness is zero.
 The normal curve is a mesokurtic curve.
 Mean deviation about mean is 4
5
of the standard deviation.
Question: A sample of 100 dry battery cells tested to find the length of life produced the results
  12 hrs and   3 hrs. Assuming the data to normally distributed, what percentage of the battery cells
are expected to have life
a) More than 15 hours.
(Ans: 15.87% )
b) Less than 6 hours.
(Ans: 2.28% )
c) Between 10 and 14 hours.
(Ans: 49.72% )
Question: In a normal distribution, 31% of the items are under 45 and 8% are over 64 . Find the mean
and standard deviation of the distribution.
`
(Ans:   50 ,   10 )
Question: If the skulls are classified as A , B and C according as length index is under 75 , between 75
and 80 or over 80 , find approximately the mean and standard deviation of a series in which A are 58% ,
B are 38% and C are 4% .
(Ans:   74.35 ,   3.23 )
Question: The incomes of a group of 10,000 persons were found to be normally distributed with mean
equal to Rs. 750 and standard deviation Rs. 50. What is the lowest income among the richest 250?
(Ans: Rs. 848)
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