PROBABILITY
.
PROBABILITY
Points to Remember:
1.
Set of all the possible outcomes in a random experiment is its Sample Space.
2.
Every cub set of a sample space is an event.
3.
Two events A and B are said to be mutually exclusive events if A  B = 
4.
A1 , A2, A3, ………………… An are mutually exclusive and exhaustive events if
n
Ai  Aj    i  j  and  Ai  S1 , where 0  P  A  1
i 1
5.
If A is an event of an experiment and S is the sample space, then the probability of A i.e. P(A) =
n(A)/n(S), where 0  P(A)  1
6.
Two events A and B are independent if P(A  B) = P(A).P(B)
7.
P(A  B) = P(A or B) = P(A) + P(B) – P(A  B)
8.
If A and B are mutually exclusively events then P(A  B) = P(A) + P(B)
9.
P(A  B  C)=P(A) + P(B) + P(C)– P(A  B)–P(B  C)– P(A  C) + P(A  B  C).
10.
If A, B, C are pairwase mutually exclusive events then
P (A  B  C) = P(A) + P(B) + P(C)
11.
P(A) + P(A’) = 1
12.
P(A/B) = P(A  B)/P(B) and P(B/A) = P(A  B)/ P(A)
General Question
Three coins are tossed simultaneously. List the sample space for the event.
Find the probability that a leap contains 53 Sundays ?
Two dice are thrown together. What is the probability that the sum of the two faces is divisible by
3 or 4.
In a group there are 3 men 2 women, 3 persons are selected at random from this group. Find the
probability that 1 man and 2 women or 2 men and 1 woman are selected.
Two balls are drawn in succession without replacement from a bag containing 10 white and 15
black balls. What is the probability that first is white and second is black.
Find the probability distribution of doublets in three throws of pair of a dice.
A random variable x has the following probability distribution:
X
0
1
2
3
4
5
6
7
P(x)
0
k
2k
2k
3k
k²
2k²
7k² + k
Graphics By:- Pradeep
1
Written By:- Raj Kumar Badhan
PROBABILITY
.
i)
Determine
K
(ii)
P(X < 3)
(iii)
P (X > 6)
(iv)
P(0<X<3)
Two persons A and B throw a die alternately till one of them gets a six and wins the game, if starts
the game. Find the respective probability of winning.
The probability that A can solve a problem is
1
2
and that of B is . If A and B both try to solve
5
3
the problem; find probability that :
i)
Only A solves the problem
ii)
One of them solve the problem
iii)
Problem is solved
GRADED QUESTIONS
LEVEL – 1
1.
Two dice are thrown simultaneously. Find the probability of getting 3 as the sum.
2.
Find the probability that a leap a year, selected at random will contain 53 Sundays.
3.
Find the probability of finding the sum 9 or 11 in a single throw of two dice.
4.
Events A and B are independent. Find P(B) if P(A) = 0.35 and P(A  B) = 0.6
5.
From a set of 17 cards, numbered 1, 2, 3 ………..17, one is drawn at random. Show that the
chance that this number is divisible by 3 or 7 is
7
.
17
6.
Two dice are thrown simultaneously. Find the probability of getting a multiple of 3 as the sum.
7.
Two dice are tossed once. Find the probability of getting an even number on first dice or getting a
total of 8.
8.
A problem in Maths is given to three students whose chances of solving it are
1 1 1
, , respectively.
2 3 4
What is the probability that the problem is solved.
9.
A speaks truth in 60% cases and B in 90% cases. In what % of cases they are likely to contradict
each other in stating the same fact ?
10.
A die is rolled, if the outcome is an even number. What is the probability that it is a prime
number?
LEVEL – 2
1.
A bag contains 20 tickets bearing numbers 1 to 20. Two tickets are drawn at random from it. Find
the probability that both the numbers on the tickets are prime.
Graphics By:- Pradeep
2
Written By:- Raj Kumar Badhan
PROBABILITY
.
2.
Three coins are tossed simultaneously. List sample space for the event.
3.
A bag contains 8 red, 3 white and 9 blue balls. Three balls are drawn at random from the bag.
Determine the probability that none of the balls drawn is white.
4.
Two balls are drawn at random from a bag containing 2 white, 3 red, 5 green and 4 black balls
one by one without replacement. Find the probability that both the balls are of different colours.
5.
Two unbiased dice are thrown. Find the probability of getting a sum less than 6.
6.
Two unbiased dice are thrown. Find the probability of getting a sum more than 7.
7.
Find the probability of getting a sum as a prime number when two dice are thrown together.
8.
A committee of 5 principals is to be selected for a group of 6 gents principals and 8 lady principals.
If the selection is made randomly, find probability that there are 3 lady principals and 2 gents
principals.
9.
Three bags contain 5 white 8 red, 7 white 6 red and 6 white 5 red balls respectively. One ball is
drawn at random from each bag. Find the probability that all the three balls drawn are of the
same colour.
10.
There are 2 red and 3 black balls in a bag. 3 balls are taken out at random from the bag. Find the
probability of getting 2red and 1 black ball or 1 red and 2 black balls.
LEVEL – 3
1.
The probability that a student A can solve a question is 6/7 and another student B solving a
questions is ¾. Assuming that the two events “A can solve the question” and “B can solve the
question” are independent, find the probability that only one of them solves the problem.
2.
One card is drawn from a well shuffled pack of 52 cards. If E is the event “the card drawn is a
king or queen” and F is the event “ the card drawn is a queen or an ace”. Then find the
conditional probability of event (E/F).
3.
A coin is tossed thrice and all eight outcomes are assumed equally likely. Let the event E be “the
first throw results in a head” and the event F be “the last throw results in a tail”. Are E and F
independent?
4.
Cards are numbered 1 to 25. Two cards are drawn one after the other. Find the probability that
the number on one card is a multiple of 7 and on the other a multiple of 11.
QUIZ
(Probability)
Two cards are drawn successively with replacement from a well shuffled Pack of 52 cards. The
probability of drawing two kings is
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3
Written By:- Raj Kumar Badhan
PROBABILITY
.
a)
1 1

52 51
(b)
1 1

13 13
(c)
1 1

13 17
(d)
1 4

13 51
Four letters are written to different persons, and address on four envelopes are also written,
without looking the Probability that no letter goes into right envelops is
a)
1
27
b)
3
8
(c)
1
9
(d)
None of these.
If A and B are arbitrary events then
a)
P(A  B)  P(A) + P(B)
b)
P(A  B)  P(A) + P(B)
c)
P(A  B) =P(A) + P(B)
d)
None of these.
The probability that at least one of the event A and B occurs is 0.6. If A and B occur
simultaneously with Probability 0.2 then P(A) + P(B) is
a)
0.4
b)
0.8
c)
1.2
d)
None of these
A die thrown 3 times and sum of 3 numbers thrown is 15 then the chance that the first throw was
a four is
a)
3
8
b)
1
18
c)
1
d)
None of these
If A and B are mutually exclusive events then P(A  B) will be equal to
a)
P(A) + P(B) b)
P(A) P(B)
c)
P(A) – P(B)
d)
P  A
P  B
A purse contains 4 copper coins and 3 silver coins, the second purse contains 6 copper coins and 2
silver coins. A coin is taken out of any purse, the probability that it is a copper coin is
a)
4
7
b)
3
4
c)
If A and B are two events P(A  B) =
3
7
d)
57
.
56
5
1
1
and P (A  B) = , P(B) =
then the events A and B are
6
3
2
a)
Dependent
b)
Independent
c)
mutually exclusive
d)
none of these.
On a toss of two die, A throw a total of 5 then the Probability that he will throw another 5 before
throws 7 is
a)
1
9
b)
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1
6
c)
4
2
5
d)
5
36
Written By:- Raj Kumar Badhan
PROBABILITY
.
A number is chosen at random among the first 120 natural numbers. The probability of the
number chosen being a multiple of 5 or 15 is
a)
1
5
b)
1
8
c)
1
6
d)
none of these.
The probability of 3 mutually exclusive events A, B, C are P(A) =
2
1
1
P(B) =
P(C) = .
3
4
6
Is the statement
a)
True
b)
wrong
c)
could be either
d)
do not know
The chance of throwing a total of 3 or 5 or 11 with two dice is
a)
5
36
b)
1
9
c)
2
9
d)
19
39
If A and B are two events such that P(A)>0 and P(B)  1 then P(  P / B  is equal to
a)
1 – P(A/B)
b)
1–
P(A)
P(B)
c)
1– P(A  B)
P B
d)
P(A)
P(B)
ASSIGNMENT
(Probability)
Four digit numbers are formed by using the digits 1, 2, 3, 4 and 5 without repeating any digit.
Find the probability that a number chosen at random is an odd number.
For two independent events A and B P(A) = 0.38 and P(A  B) = 0.69. Find P(B).
Three dice are thrown together. Find the probability of getting a total of at least six.
If P(E) = 0.45, P(F) = 0.55, P(E  F) = 0.75. Find
i)
P(E  F)
ii)
P(E/F)
In a single throw of two dice, determine the probability of not getting the same number on the two
dice.
A bag contains 2 white balls, 3 black balls, 4 red balls and 5 green balls. If two balls are drawn at
random one by one without replacement from the bag, the probability that the balls drawn are of
different colours.
If the odds in favour of an event are 4 to 5, find the probability that it will occur.
In a single throw of two dice, find the probability of getting total of 9 or 11.
Two cards are drawn successively with replacement, from a well shuffled pack of 52 cards. Find th
probability distribution of the number of aces.
Graphics By:- Pradeep
5
Written By:- Raj Kumar Badhan
PROBABILITY
.
Out of eleven tickets marked with number 1 to 11 three tickets are drawn at random find the
probability that the numbers on them are in A.P.
11.
A pair of dice is tossed together. Find the probability that the sum is either 5 or 7.
Find the probability that in a random arrangement of the letters of the word “MATHEMATICS”
the consonants occur together.
In a class 30% of the students offered Mathematics, 20% offered Chemistry and 10% offered
both. If a student is selected at random, find the probability that he has offered Mathematics or
Chemistry.
1 1 1
, , what
2 3 4
is the probability that the problem is solved? Also find the probability that only one of them solve
it correctly.
A problem in Mathematics is given to three students whose chances of solving it are
2
5
and of student B passing is . Assuming two
9
9
events: “A passes”, “B passes” as independents. Find the probability of :
The probability of student A passing an exam is
i)
Only A passing the examination
ii)
Only one of them passing the examination
EVALUATION
(Probability)
1.
Two cards are drawn from a well-sluffed pack of 52 cards one after the other without
replacement. Find the probability that one of these is a queen and other is king of opposite colour.
2.
A bag contain 2 white and 4 black balls while another bag contains 6 white and 4 black balls. A
bag is selected at random and a ball is drawn. Find the probability that the ball thrown is of white
colour.
3.
The probability that a company executive travel by plane is 2/3 and that he will travel by train is
1/5. Find the probability of his travel by plane or train.
4.
In a simultaneous toss of two coins, find the probability of getting 2 heads. What is the probability
of getting ‘exactly 1 head ?
5.
Two person A and B throw a pair of dice alternately beginning with A. Find the probability that B
gets a doubled and wins before A gets a total of 9 to win.
6.
A bag contains 4 yellow and 5 red balls and another bag contains 6 yellow and 3 red balls. A ball is
drawn from the first bag and without seeing its colour, it is put into the second bag. Find the
probability that if now a ball drawn from the second bag, it is yellow in colour.
7.
Two dice are rolled once. Find the probability that the number on the two dice are different. What
is the probability that total is at least 4 ?
Graphics By:- Pradeep
6
Written By:- Raj Kumar Badhan
PROBABILITY
.
8.
Three balls are drawned one by one without replacement from a bag containing 5 white and 4 red
balls. Find the probability distribution of the number of red balls drawn.
9.
There are 3 red and 2 black balls in a bag. 3 balls are taken out at random from the bag. Find the
probability of getting 2 red and 1 black or 1 red and 2 black balls.
10.
A card is drawn from a pack of cards. Find the probability of getting a king or a heart or a red
card.
11.
A bag contains 3 white, 3 black and 2 red balls. One by one, three balls are drawn without
replacing them. Find the probability that the third ball is red.
Graphics By:- Pradeep
7
Written By:- Raj Kumar Badhan