PROBABILITY
***********************************************************************************************
Def 1: If A and B are any two events associate with a random experiment, then conditional probability of B when A has already
occurred, written as P(B/A) is defind as P(A
 B)/P(A).
Multiplication theorem of Probability
If A and B are two events, then
P(A/B) = P(A  B)/ P( B)
Independent Event : Two event A and B are independent iff P(A
 B ) = P(A).P(B)
The law of total probability
If E , E , E ,…………..,E be a set of Mutually exclusive and exhaustive events associated with a random experiment and E be any
event associated with the same experiment, then
P( E) =

P ( E/ E ).P( E ).
Bayes Theorem
If E , E , E ,……….,E form a set of mutually exclusive and exhaustive events and E is any event associated with the same
experiment, then P( E / E ) = P (E / E ).P ( E ) /

P ( E / E ) P ( E ).
def 2: Random variable
A random variable is a real valuaed function X defined on the sample space S of a random experiment wihich associates a uniqe
real number X(
 ) to each   S.
So, Domain of X= S; Range Of X = { x(
Mean(
 )= E( x ) =
Varience (


p.x
) = E ( x ) – ( E (X) )
Standard deviation =
Variance.
 ) :  S}.
1)A card is drawn from a well shaffled pack of 52 cards. The outcome is noted and the
pack is again reshuffled without replacing the card. Another card is then drawn. What is
the probability that a first card is a spade and the second is a black king
2)The probability of student A passing an examination is 3/5 and of student B passing is
4/5. Assuming the two events: A passes, B passes , as independent find the probability
of :
a) both student passing the examination.
b) only A passing the examination.
c) only one of the two passing the examination.
d) neither of the two passing the examination.
3)A bag contains 4 white and 6 black balls. One ball is drawn and laid aside without
noticing its colour. Another ball is then drawn. What is the probability that the second
ball is black?
4)Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by
one without replacement. What is the probability that
a)one is white and one is black.
b)at least one ball is red ?
5)A problem in Mathematics is given to three student whose chances of solving it are
1/3 , 1/5 , 1/6. What is the probability that atleast one of them solves the problem?
6)An urn contains 25 balls numbered 1 to 25. Suppose an odd number is considered a
success . Two balls are drawn from the urn with replacement. Find the probability of
getting
a)two successes
(b) exactly one success (c) at least one success (d) no success.
7) A speaks truth in 60% cases and B in 90% cases. In what percent of cases are they
likely to contradict each other in stating the same fact?
8)Two persons throw a alternately till one of them get a three and wins the game. Find
their respective probabilities of winning.
9)A and B are two independent events. The probability that both A and B occur is 1/6
and the probability that neither of them occurs is 1/3. Find the probability of the
occurrence of A.
10tree persons A, B and C throw a dice in succession till one get a six and wins the
game. Find their respective probabilities of winning.
11)A company has two plants to manufacture scooters. Plant-1 manufactures 70% of
the scooters and plant-2 manufactures 30%. At plant -1, 80% of the scooters are rated
of standard quality and at plat -2 , 90% of the scooters are rated as standard quality. A
scooter is chosen at random and is found to be standard quality. Find the probability
that it has come from plant -2.
12)A pack of playing cards was found to contain only 51 cards which are examined are
all red, what is the probability that the missing card is black.
13A bag P contains 3 white and 4 red balls and a bag Q(identical with P) contain 5
white and 7 red balls. One ball is drawn at random from a bag selected at random. If the
ball drawn is found to be red, find the chance that it is drawn from the bag Q.
14)A factory has three machines X, Y and Z producing 1000,2000 and 3000 bolts per
day respectively. The machine X produces 1% defective bolts , Y produces 1.5% and Z
produces 2% defective bolts. At the end of day, a bolt is drawn at random and is found
defective. What is the probability that the defective bolt is produce by the machine X?
15)Given three identical boxes I, II and III each containing two coins. In box I, both coins
are gold coins, in box II, both are silver coins and in the box III, there is one gold and
one silver coin. A person chooses a box at random and take out a coin . If the coin is
gold, what is the probability that the other coin in the box is also of gold?
16)A man is known to speak truth 3 out of 4 times . He throws a die and reports that it is
a six. Find the probability that it is actually a six.
17)A bag contains 4 balls. Two balls are drawn at random, and are found to be white.
What is the probability that all the balls are white.
18) A family has two children. Find the probability that both are boys, if it is known that
i) at least one of the children is a boy.
ii) the elder child is a boy.
19)A purse contains 2 silver and 4 copper coins. A second purse contains 4 silver and 3
copper coins. If a coin is pulled at random from one of the two purses, what is the
probability that it is a silver coin ?
20)Find the probability of drawing one rupee coin from a purse with two compartments
one of witch contains 3 fifty paise coins and 2 one rupee coins and other contains 2 fifty
paise coins and 3 one rupee coins.
21)A dice is tossed twice. A “success” is getting an even number on a toss. Find the
probability distribution of the number of successes. Also draw the graph of this
probability distribution.
22) Two cards are drawn in succession from a well shuffled deck of 52 cards, the first
card being replaced, before the second is drawn. Let X denotes the number of spades
drawn. Find the probability distribution of X.
23)Find the probability distribution of the random variable X which denotes the number
of times “a total of 8” appears in two throws of a pair of dice.
24)A box contains 12 bulbs of which 3 are defective. A sample of 3 bulbs is selected
from the box. Let X denote the number of defective bulbs in the sample, find probability
distribution of X.
25)Find the probability distribution of number of doublets in four throws of a pair of dice.
26)Find the mean and variance of the random variable X , where probability distribution
is given by the following table:
X
-2
-1
0
1
2
P(X)
.1
.2
.3
.2
.15
3
.05
27)Find the mean and standard deviation of the probability distribution of the number
obtained when a card is drawn at random from a set of 7 cards numbered 1 to 7 .
28)A dice is thrown thrice. Find the mean and variance of the number of times a “six” is
obtained.
29)Find the probability distribution of the number of heads when three coins are tossed.
Also find the mean number of heads in the above case.
30)A pair of dice is thrown twice .Let X denote the number of times , “a total of 9 is
obtained”. Find the mean and variance of the random variable X