Tutorial – 1
Lecture by: N.V.Nagendram
--------------------------------------------------------------------------------------------------------------Problem 1: If the letters of the word “REGULATIONS” be arranged at random, what is the
 6
chance that there will be exactly 4 letters between R and E?
Ans  
 55 
Solution: total number of letters = 11 for 2 letters R, E exhaustive cases of ways
11
P2=11X10= 110
Favourable cases that there will be exactly 4 letters between R and E = 2 X 6 = 12 ways
2X 6
12
6


 P(4 letters b/w R & E) =
. Hence the solution.
11X 10 110 55
LBRCE\S&H\P AND S\TUTORIAL\NVN\19102013
Problem2: Find the probability that four S’s come consecutively in the word
 4 
“MISSISSIPPI”? 

 165 
Solution: Total number of letters in MISSISSIPPI = 11 permutations are 4 are of one kind
(S), 4 of other kind ( I) , 2 of a third kind ( P ) and 1 of fourth kind ( M )
so exhaustive case =
11!
4! 4! 2! 1! 1!
after arrangement of SSSS 4 S’s remaining are 7 letter MIIIPPI i.e., of 4 of one kind (I) , 2 of
7!
other kind (P) and 1 of fourth kind (M) so favourable case =
4! 2! 1! 1!
11!

 4! 4! 2! 1! 1! 8 X 7! X 4!
4

 Probability of (4S’s consequently in Mississippi) = 

11!
165
 7!

 4! 2! 1! 1!


Hence the solution.
Problem 3: From 25 tickets, marked with the first twenty five numerals, one is drawn at
 8 2
random. Find the chance that (i) it is multiple of 5 or 7. (ii) It is multiple of 3 or 7.  , 
 25 5 
Solution: from 25 tickets as sample space and consider in first twenty five numerals as
exhaustive case = 25.
( i) multiples of 5 are 5,10,15,20,25 = 5 numbers and multiples of 7 in 25 = 7,14,21 = 3
numbers total favourable cases = 5 + 3 =8
53 8

P(E)=
25
25
( ii) similarly multiples of 3 are 3,6,9,12,15,18,21,24 = 8 and multiples of 7 = 7,14,21
But one number 21 is common in both so , favourable = 8 + 3 -1 = 10
8  3  1 10 2

 .
25
25 5
Problem 4: Among the digits 1, 2, 3, 4, 5 at first one is chosen and then a second selection is
made among the remaining four digits. Assuming that all twenty possible outcomes have
equal probabilities, find the probabilities that an odd digit will be selected (i) the first time ,
(ii) the second time and (iii) both the times.
P( E )=
Solution: Exhaustive cases = 20
First time: (1,2)(1,3)(1,4)(1,5); (3,1)(3,2)(3,4)(3,5) ;(5,1)(5,2)(5,3)(5,4) = 12
12 3

P( E =first time ) =
20 5
( ii) second time: (2,1)(3,1)(4,1)(5,1); (1,3)(2,3)(4,3)(5,3); (1,5)(2,5)(3,5)(4,5) = 12
12 3

P(E = second time) =
20 5
( iii) third both the times: (1,3)(1,5);(3,1)(3,5);(5,1)(5,3) = 6
6
3

P(E = third both the times) =
Hence the solution.
20 10
Problem 6: What is the probability of getting 9 cards of the same suit in one hand at a game
of bridge ?
Solution: Exhaustive cases =
52
C13
Number of ways in one hand getting 9 cards of one suit = 13C9 remaining 52 – 13 = 39 other
suit of four i.e., 4 X 39C4
So, total favourable cases = 4 X
P( E ) =
4X
39
C4 X
52
39
C4 X 13C9
13
C9
C13
 0.00037 hence the solution.
Problem 7(a): Two cards are drawn at random from a well shuffled pack of 52 cards. Show
1
.
that the chance of drawing two aces is
221
Solution : From a pack of 52 cards 2 cards can be drawn in 52C2 ways all being equally
likely. So, exhaustive number of cases = 52C2.
In a pack there are 4 aces and therefore 2 aces can be drawn in 4C2 ways.
4
P( E ) =
C2
1

.
C 2 221
52
Problem 7(b): From a pack of 52 cards, three are drawn at random. Find the chance that they
are a king, a queen and a knave.
Solution
: Exhaustive number of cases = 52C2.
A pack of cards contains 4 kings, 4 queens and 4 knaves.
A king , a queen and a knave can each be drawn in 4C1 ways and since each
way of drawing a king can be associated with each of the ways of drawing a queen and a
knave, the total number of favourable cases 4C1 X 4C1 X 4C1.
 P( E ) =
4
C1 X 4C1 X 4C1
4X 4X 4
16
.


52
52 X 51X 50 5525
C3
Problem 7(c): Four cards are drawn from a pack of cards. Find the probability that
(i) all are diamond (ii) there is one card of each suit and
(iii) there are two
spades and two hearts.
Solution
: Exhaustive number of cases = 52C4.
13
P( E1 ) =
13
C1 X 13C1 X 13C1 X 13C1
C4
=
0.0026
and
P
(
E
)

 0.1054
2
52
52
C4
C4
13
C2 X 13C2
P(E3 ) 
 0.0224 .
52
C4
Problem 8: A bag contains 3 red, 6 white and 7 blue balls. What is the probability that two
balls drawn are white and blue?
Solution : Total number of balls = 3 + 6 + 7 = 16.
16!
Now out of 16 balls, 2 can be drawn in 16C2 =
= 120
(14!)2!
Out of 6 white balls 1 ball can be drawn in 6C1 ways and out of 7 blue balls 1 ball
can be drawn in 7C1 ways.
Since each of the former cases can be associated with each of the latter cases,
42
total number of favourable cases is 6C1 X 7C1 = 6 X 7 = 42. Therefore, P( E ) =
.
120
Problem 9: What is the chance that a leap year selected at random will contain 53 Sundays?
Solution : In a leap year there are 52 weeks and 2 consecutive extra days.
The following are the possible combinations for these two extra days:
Sunday
&
Monday
Monday
&
Tuesday
Tuesday
Wednesday Thursday Friday
&
&
&
&
Wednesday Thursday
Friday
Saturday
Saturday
&
Sunday
In order to that leap year selected at random should contain 53 Sundays one of the
two extra days must be Sunday. Since out of the above 7 possibilities 2 i.e., (i) and
2
(vii) are favourable to this event.  P( E ) = .
7
Hence the solution.
Tutorial – 7
Lecture by: N.V.Nagendram
-------------------------------------------------------------------------------------------------------------Problem 1: A factory has three production lines I,II and III contributing 20%, 30% and 50%
respectively, to its total output. The percentages of substandard items produced by lines I,II
and III are respectively. 15, 10 and 2. If an item chosen at random from the total output is
found to be substandard , what is the probability that the item is from line I?
LBRCE\S&H\P AND S\TUTORIAL\NVN\22102013
Solution: We first represent the problem as below diagram:
E3
0.02
A
0.5


0.3E2
0.10
B



0.2
E1
0.15
C


By Bayes formula
P( E1 \ A) 
P( E1 ) . P( A \ E1 )
3
 P( E ) P( A \ E )
i 1
i

0.2 X 0.15
 0.069
0.2 X 0.15  0.3 X 0.10  0.5 X 0.02
i
Hence the solution.
Problem 2: Two similar Urns, A,B contain 2 white and 3 red balls, 4 white and 5 red balls
respectively. If a ball is selected at random from one of the Urns, then find the probability
that the Urn is B, when ball is red?
Solution:
Urn
A
B
White balls
2
4
Red balls
3
5
The events of selecting the Urns are equally likely.
Let the event of selecting the first Urn A be E1 and the second Urn B be E2.
1
3
 P(E1) = P(E2) = ; P(R \E1) = Probability of drawing a red ball from the first urn =
2
5
5
P(R \E2) = Probability of drawing a red ball from the second urn = .From Bayes theorem
9
1
5
X
P( E ) . P( R \ E1 )
25
2
9
P( E 2 \ R)  3 2


3 1
5
52
1
P ( Ei ) P ( R \ Ei )  X  X 

5 2
9
2
i 1
Hence the solution.
Objective Type Practice Test – 3 LBRCE
Probability & Statistical Applications (I-MCA)
By N V Nagendram
1.) A bag contains 2 white and 3 black balls. Four persons A,B,C and D in the order named
each draw one ball and do not replace it. The person to draw a white ball receives Rs.200/Determine their expectations.
[Ans. Rs.80,Rs.60,Rs.40, Rs.20]
2.) There are three alternative proposals before a business man to start a new project:
Proposal
A
Description
Profit of Rs.5,00,000/- with a probability of 0.6 or a loss of
Rs.80,000/- with a probability of 0.4
B
Profit of Rs.10,00,000/- with a probability of 0.4 or a loss of
Rs.2,00,000/- with a probability of 0.6
C
Profit of Rs.4,50,000 with a probability of 0.8 or a loss of
Rs.50,000/- with a probability of 0.2.
If he wants to maximise the profits and minimize the loss, which proposal he should prefer.
[Ans. Proposal C]
3.) A dealer in refrigerators estimates before from his past experience the probabilities of his
selling refrigerators in a day. These are as follows:
Number of
0
1
2
3
4
5
6
Refrigerators
sold in a day
Probability
0.03
0.20
0.23
0.25
0.12
0.10
0.07
Find the mean number of refrigerators sold in a day?
[Ans. 2.81]
4.) The probability that a man fishing at a particular place will catch 1,2,3,4 fish are 0.4, 0.3,
0.2 and 0.1 respectively. What is expected number of fish caught?
[Ans. 2]
5.) When 2 dice are rolled find the probability that the sum of the dots on the two faces is
even number less than 6?
1
[Ans. ]
9
1
6.) A, B are 2 independent events such that the probability of both the events to occur is
6
1
and the probability of both the events do not occur is
. Find the probability of A.
3
1 1
[Ans. or ]
2 3
7.) In a group consisting of equal number men and women 10% of the men and 45% of the
women are unemployed. If a person is randomly selected from the group then find the
29
probability that the person is an employee.
[Ans.
]
40
8.) If P(A  B)= 0.65, P(A  B) = 0.15, then find the value of P( A ) + P( B ).
[Ans.1.2]
9.) Mr. A is called for 3 interviews. There are 5 candidates at the first interview, 4 at the
second and 6 at the third. If the selection of each candidate is equally likely then find the
1
probability that A will be selected for at least one post.
[Ans. ]
2
10.) A box contains 3 red and 7 white balls. One ball is drawn at random and in its place, a
ball of the other colour is placed in the box. Now if one ball is drawn at random from the box
then find the probability that it is a red ball.
[Ans. 0.34]
11.) The probability that a boy will get a scholarship is 0.9 and that a girl will get is 0.8. What
is the probability at least one of them will get the scholarship?
[Ans. 0.98]
12.) A committee of 6 is to be formed from 7 men and 4 women calculate the probability that
5 53
the committee will consist of (i) exactly 2 women (ii) at least 2 women.
[Ans. ; ]
11 66
13.) If 7 cards are drawn at random, from a pack of 52 cards then find the probability that 3 of
26
them are red and 4 are black?
[Ans.
C3 X 26C 4
]
52
C7
14.) A,B ,C are three routes from the house to the office. On any day, he route selected by the
officer is independent of the climate. On a rainy day, the probabilities of reaching the office
1 1 1
, , respectively. If a rainy day, the officer is late to the
late, through these routes are
25 10 4
10
office then find the probability that the route to be B.
[Ans.
]
39
15.) Three boxes numbered, I,II and III contain balls as follows:
Box
White
Black
Red
I
1
2
3
II
2
1
1
III
4
5
5
One box is randomly selected and a ball is drawn from it. If the ball is red, then find the
1
probability that it is from box II.
[Ans. ]
4
1
. If
3
India and England play, 3 matches, what is the probability that, (i) India will lose all the three
8 1
matches
(ii) India will win at least one match?
[Ans. , ]
9 9
17.) If one card is drawn at random from a pack of well shuffled cards then find the
4 X 3X 2 X1
probability that all the 4 numbers to be aces.
[Ans.
]
52 X 51X 50 X 49
18.) When 3 dice are rolled, find the probability, that the sum on the three dice is 6 or less?
5
[Ans.
]
54
19.) Six boys and six girls sit in a row. Find the probability of (i) all the girls sit together (ii)
16.) The probability that India wins a cricket match against England is given to be
7!6! 2(6!) 2
all the girls sit together and also boys sit together ?
[Ans.
]
;
12! 12!
20.) A, B, C are aiming to shoot a balloon. A will succeed 4 times out of 5 attempts. The
chance of B to shoot the balloon is 3 out of 4 and that of C is 2 out of 3. If the three aim the
balloon simultaneously, then find the probability that at least two of them hit the balloon.
5
[Ans. ]
6
21.) A college is divide into 2 groups. The first group consists of 5 science and 3 engineering
subjects, where as the second group consists of 3 science and 5 engineering subjects. A die is
rolled, a subject from the first group is selected if the die shows 3 or 5 and otherwise a subject
from the second group is selected. Find the probability of selecting the engineering subject?
13
[Ans.
]
24
22.) There are 3 black and 4 white balls in one bag. 4 black and 3 white balls in the second
bag. A die is rolled and the first bag is selected if it is 1 or 3, and the second bag for the rest.
11
Find the probability of drawing a black ball, from the bag, thus selected?
[Ans.
]
21
1 1 1
23.) The probabilities of 3 students to solve a problem in mathematics are
, ,
2 3 4
3
respectively. Find the probability that the problem to be solved?
[Ans. ]
4
24.) If E1, E2, E3 are independent events are as follows:
1
1
1
P( E1  E 2  E3 )  , P ( E1  E 2  E3 )  , P( E1  E 2  E3 )  , then find P(E1), P(E2) and
4
8
4
1 1 1
P(E3)?
[Ans. , , ]
2 3 4
25.) There are 7 red and 3 white marbles in an urn. If 3 marbles are randomly drawn from I,
one after another, then find the probability for the first 2 being red and the third being white?
7
*** *** ***
[ans.
]
40
Objective Type Practice Test – 2 LBRCE
Probability & Statistical Applications (I-MCA)
By N V Nagendram
01. The dice are rolled simultaneously. The probability of getting 12 spots is _______
1
1
1
02. Given that P(A) = , P(B) = , P(A/B) = , the probability (B/A) is equal to ____
3
4
6
03. Also P(B/ A ) for problem 2 is ________________
04. A number is selected randomly from each of the two sets
1,2,3,4,5,6,7,8
2,3,4,5,6,7,8,9 then the probability that the sum of the numbers is equal to 9 is__
05. In tossing three coins at a time, the probability of getting atmost one head is____
06. For any two events A and B, P (A – B ) is equal to ______________
07. Out of 20 employees in a company Five are graduates. Three employees are selected at
random. The probability of all three being graduates is ____________
1
3
11
08. Given that P(A) = , P(B) = , and P(A  B) =
then P(B/A) = _____________
3
4
12
09. A coin is tossed six times. The probability of obtaining heads and tails alternately is
___________________
10. What pair of dice thrown at a time, the probability of getting a sum more than that of 9 is
_____________
11. An urn contains 5 yellow, 4 black and 3 white balls. Three balls are drawn at random. The
probability that no black ball is selected is _____________________
12. Four cards are drawn from a pack of 52 cards. The Probability that out of 4 cards being 2 red
and 2 black is ___________________
13. The probability of throwing an odd sum with two fair dice is _________________
14. Out of 20 employees in a company, five are graduates. Three employees are selected
at random. The Probability of all the three being graduates is ___________________
15. One of the two events much happen; given that the chance of one is
1
of the other. The odd
4
in favour of the other is ________________
16. If A tells truth 4 times out of 5 and B tell truth 3 times out of 4. The probability that both
expressing the same fact contradict each other is _______________________________
17. In answering a question on a multiple choice test, an examinee either knows the answer with
probability p or the guess with probability (1 – p). Let the probability of answering the
1
question correctly be 1 for an examines who knows the answer and
who guesses ( m
m
being the number of the multiple choice alternatives)
18. There are two bags one bag contains 4 red and 5 black balls and the other 5 red and 4 black
balls. One ball is to be drawn from either of the two bags. The probability of drawing a black
ball is _____________________
19. A machine part is produced by three factories A, B and C. Their proportional production is
25, 35 and 40% respectively. Also the % defectives manufactured by three factories are 5, 4
and 3 respectively. A part is taken at random and is found to be defective. The probability
that selected part belongs to factory B is ______________
20. The chance that doctor A will diagnose a disease X correctly is 60%. The chancethat a patient
will die by his treatment after correct diagnosis is 40% and the chance of death by wrong
diagnosis is 70%. A patient of doctor A, who had disease X, died. The probability that his
disease was diagnosed correctly as ________________________
21. A can hit a target 2 times in 5 shots. B 3 times in 5 shots and C 4 times in 5 shots. They fire a
valley (each try once to hit the target). The probability that two shots hit is _____
22. If one card is selected at random from 100 cards numbered as 00,01, …,99. Suppose x and y
are the sum and product of the digits on the select card. If I is the whole number, the
probability P(x = I /y = 0)
23. There is 80% chance that a problem will be solved by a statistics student and 60 % chance is
there that the same problem will be solved by the mathematics student. The probability that at
least the problem will be solved is __________________________
24. If a bag contains 4 white and 3 black balls. Two draws of 2 balls are successively made, the
probability of getting 2 white balls at first draw and 2 black balls at second draw when the
balls drawn at first draw are replaced is _____________________________
25. In a city 60% read newspaper A, 40% read newspaper B and 30% read newspaper C, 20 %
read A and B, 30% read A and C, 10% read B and C. Also 15% read papers A, B and C. The
% of people who don’t read any of these newspapers is ______________________
**********
***************
*********
KEY PracticeTest-2
LBRCE\S&H\NVN\PS-27062013
1. 25/215
2.
1/8
3. 5/16
4. 7/64
5.
½
6. P(A) – P(AB)
7. 1/ 114
8.
1 /2
9. 1 / 32
10. 1/6
11. 7/55
12. 650/833
13. ½
14. 1/114
15. 1 : 4
16. 7/20
17. mp/[1+(m-1)p]
18. 1/2
19. 4 /11
20
21. 58 /125
22. 19 /100
23. 0.92
25. 15%.
6 / 15
24. 2 / 49
Objective Type Practice Test – 1 LBRCE
Probability & Statistical Applications (I-MCA)
By N V Nagendram
1. A box contains 3 black and 5 white balls. If a ball is drawn at random what is the
probability that it is (i) black (ii) White ?
2. If a page is randomly selected from a book of 100 pages, then find the probability that the
sum of the digits of the page is 10?
3. When 2 dice are rolled simultaneously find the probability for the sum on the two faces
will be 10?
4. When 4 coins are tossed simultaneously, find the probability to get 2 heads and 2 tails?
5. When 3 coins are tossed simultaneously, find the probability to get at least one head?
6. Find the probability, that a non-leap year contains (i) 53 Sundays (ii) 52 Sundays?
7. If a card is drawn from a pack of cards, then find the probability for the card to be (i) an
ace (ii) a club?
8. If 6 cards are drawn at random, from a pack of cards, then find the probability to get 3 red
and 3 black cards?
9. If a number is selected at random from the natural numbers 1 to 10, then fnd the
probability for the number to be (i) prime number (ii) a perfect square?
10. If 3 English , 4 Telugu and 5 Hindi books are arranged in a shelf in one row then find the
probability that the books of the same language are side by side?
11. If 3 cards are drawn from a pack of cards, then find the probability for the cards to be a
king, a queen and a jack?
12. When 2 dice are rolled find the probability to get an even number on each die.
13. A bag contains 12 two rupee coins, 7 one rupee coins, 4 half rupee coins. If 3 coins are
selected at random, then find the probability that (i) the sum of the 3 coins is maximum (ii)
the sum of the 3 is minimum (iii) each coin is of different value ?
2 3
14. The probabilities of A and V to pass an examination are
,
. Find the probability that
10 10
only one of them, to pass the examination.
15. In an experiment of drawing a card from a pack, the event of getting a spade is denoted
by A, and getting a pictured card(King, Queen or Jack) is denoted by B. Find the probabilities
of A and B. Explain the events A  B, A  B and also find their probabilities.
16. If 2 cards are drawn from a pack of cards then find the probability for getting at least one
club?
17. In a class of 60 boys and 20 girls, half of the boys and half of the girls know cricket. Find
the probability of a person selected from the class is either a boy or a person who knows
cricket?
18. In a box containing 15 bulbs, 5 are defective. If 5 bulbs are selected from the box, at
random then find the probability of the event, that (i) None of them is defective (ii) only ne of
them is defective and (iii) At least one of them is defective?
19. A and B are seeking admission into EAMCET. If the probability for A to be selected is
0.5 and that of both to be selected is 0.3, is it possible that, the probability of B to be selected
is 0.9?
20. For any two events A and B, show that P( A  B)  1  P( A  B)  P( A)  P( B) .
21. Two persons A and B are rolling a die, on the condition that the person who gets 3, will
win the game. If A starts the game, then find the respective probabilities of A and B to win
the game?
22. A and B are rolling dice, on the condition that the person who gets a sum of 9 on both the
dice, will win the game. If A starts the game, then show that the ratio of their probabilities of
winning the game is 9 : 8.
23. A, B, C are 3 News Papers published from a city. 20% of the population read A, 16%
read B, 14% read C, 8% both A and B, 5% both A and C, 4% B and C and 2% read all the
three. Find the % of the population who read at least one paper?
24. If one ticket is randomly selected from, tickets numbered 1 to 30 then find the probability
that the number on the ticket is (i) a multiple of 5 or 7 (ii) a multiple of 3 or 5?
25. If a card is drawn from a well shuffled pack then find the probability that, the card is a
king or queen?
26. If 4 cards are drawn at a time from a pack of cards then find the probability that 4 cards to
be a king, a queen, a jack and an ace.
27. in an experiment of rolling 2 dice, find the probability that, the dots on the second die is
less than that on the first die.
2
28. The probability for a contractor to get a road contract is
and to get a building contract is
3
5
4
. The probability to get at least one contract is . Find the probability to get both the
9
5
contracts.
29. A bag contains 2 white, 3 black and 4 green balls. If 2 balls are drawn from it, one after
another then find the probability that the first one is white and the second one is black?
30. There are 2 white and 4 black balls in urn A; there are 4 white and 7 black balls. If one
ball is randomly replaced from A into B, and a ball is drawn from B then find the probability
for the ball to be a white one?
***
***
***
KEY Test – 1 LBRCE
01.
5 3
,
8 8
07.
02.
9
100
08.
1
12
3
04.
8
03.
05.
7
8
06.
1 6
,
7 7
1 1
,
13 4
26
09.
C3 X 26C3
52
C6
4 3
,
10 10
(3!) 2 (4!)(5!)
12!
64
11. 52
C3
10.
12.
1
4
By N V Nagendram
13.
19. No
4
12C 3 C3 12 X 7 X 4
,
, 23
23
C 3 23C3
C3
14.
19
50
1 3 3 11
, , ,
4 13 52 26
15
16.
34
15.
17.
7
8
18.
12 50 131
,
,
143 143 143
20. ****
6 5
,
11 11
22. 9 : 8
21.
23. 35%
24.
1 7
,
3 15
25.
26.
2
13
256
C4
52
5
12
19
28.
45
27.
29.
1
12
30.
16
39