# Class 11 Maths Ncert Solutions Ex 16.2

## Class 11 Maths Ncert Solutions Chapter 16 Ex 16.2

Question 1

An experiment consists of rolling of a die .Event E denotes that the die shows 4 and event F denotes that the die shows even number.

Are these both events that is A and B mutually exclusive?

Sol:

In an experiment when a die is rolled the sample space is expressed as = {6 , 5 , 4 , 3 , 2 , 1}

According to the question,

A= {4} and B= {6, 4, 2}

It is observed from the above mentioned sample space of E and F is

= AB$A\cap B$ = {4} = ϕ$\phi$

It is hereby observed that events A and B are not mutually exclusive events.

Question 2

An experiment consists of a die thrown in which the following events occurred:

(a) P : numbers less than 7

(b) Q : numbers larger than 7

(c) R : numbers which are product of 3

(d) S : numbers which are smaller than 4

(e) T : even numbers which are larger than 4

(f) U : numbers not less than 3

Also, PQ$P\cup Q$PQ$P\cap Q$ , QR$Q\cup R$ , TU$T\cap U$ ,

ST$S\cap T$ , P – R , S – T ,

Sol:

(a) P = {6, 5, 4, 3, 2, 1}

(b) Q = ϕ$\phi$

(c) R = {6, 3}

(d) S = {3, 2, 1}

(e) T = {6}

(f) U = {6, 5, 4, 3}

Question 3

An experiment consists throwing of a pair of dice and noting down the numbers that came up. Determine the following events:

(i) The sum of numbers is larger than 8

(ii) 2 occur on both of the die.

(iii) The sum of numbers is at least 7 and a multiple of 3.

Determine the events which are mutually exclusive?

Sol:

Sample space when a pair of dice is rolled (S) = { (x,y) : x,y = 6,5,4,32,1 }

= { (6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6)

(5,1) , (5,2)  , (5,3) , (5,4) , (5,5) , (5,6)

(4,1) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6)

(3,1) , (3,2)  , (3,3) , (3,4) , (3,5) , (3,6)

(2,1) , (2,2)  , (2,3) , (2,4) , (2,5) , (2,6)

(1,1) , (1,2)  , (1,3) , (1,4) , (1,5) , (1,6)}

P = {(4,6) , (5,4) , (3,6) , (6,4) , (6,5)  (4,5) , (5,5) , (5,6) , (6,3)  , (6,6) }

Q = {(6,2) , (5,2) , (4,2) , (3,2) ,(1,2) , (2,6) , (2,5) , (2,4) , (2,3) , (2,2) , (2,1) }

R = {(6,6) , (4,5) , (5,4) , (6,3) , (6,6)}

It is noted that:

PQ$P\cap Q$ = ϕ$\phi$

QR$Q\cap R$ = ϕ$\phi$

RP$R\cap P$ = {(6,6) , (6,3) ,(5,4) , (4,5) , (4,5) , (3,6) } =  ϕ$\phi$

Therefore, events P, Q and R are mutually exclusive.

Question 4

In an experiment 3 coins at a time .Let P be the event in which there 3 heads. Q be the event  in which there is 2 heads and 1 tail  and R be the  event in  which there is 3 tails and S denotes the event in which head shows in the first coin. Find the events:

(a) Mutually exclusive?

(b) Simple?

(c) Compound?

Sol:

Sample space of the experiment when 3 coins are tossed together.

S = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}

P = {HHH}

Q = {HHT, HTH, THH}

R = {TTT}

S = {HHH, HHT, HTH, HTT}

It is noted that:

PQ$P\cap Q$ = ϕ$\phi$

PR$P\cap R$ = ϕ$\phi$

PS$P\cap S$ = {HHH} = ϕ$\phi$

QR$Q\cap R$ = ϕ$\phi$

QS$Q\cap S$ = {HHT , HTH} = ϕ$\phi$

RS$R\cap S$ = ϕ$\phi$

(a) Events P and Q, events P and R, events Q and R and R and S are all mutually exclusive events.

(b) If an event of an experiment is having a sample point in a sample space it is known as simple event. Therefore P and Q are simple events.

(c) If an event of an experiment is having more than one sample point in a sample space it is known as compound event. Therefore Q and S are compound events.

Question 5

In an experiment 3 coins are tossed.

(a) 2 events which are mutually exclusive.

(b) 3 events which are mutually exhaustive and exclusive.

(c) 2 events, which are not mutually exclusive.

(d) 2 events which are mutually exclusive but not exhaustive.

(e) 3 events which are mutually exclusive but not exhaustive.

Sol:

Sample space when 3 coins are tossed all together = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}

(a) 2 events are said to be mutually exclusive:

Q = no tails

P = {TTT}, Q = {HHH} are disjoint

(b) 3 events are said to be mutually exclusive & exhaustive if :

R = at least 2 heads

P = {TTT}

Q = {TTH, THT, HTT}

R = {THH, HTH, HHT, HHH}

Because PQ$P\cap Q$ = QR$Q\cap R$ = RS$R\cap S$ = ϕ$\phi$ = PQR=S$P\cap Q\cap R = S$

(c) 2 events are not mutually exclusive:

Q = at least 2 heads

P = {HHH}

Q = {THH , HTH , HHT , HHH }

Because PQ$P\cap Q$ = {HHH} ≠ ϕ$\phi$

(d) Event which are not exhaustive but are mutually exclusive:

Q = exactly 1 tail

P = { TTH , THT , HTT }

Q = {THH , HTH ,HHT }

Because PQ$P\cap Q$ = ϕ$\phi$

But, PQ$P\cup Q$ ≠ S

(e) 3 events that are not exhaustive but they are mutually exclusive:

Q = 1 head and 2 tails

R = 1 tail and 2 heads

P = { HHH }

Q = { TTH , THT , HTT }

R = { THH , HTH , HHT }

Because PQ$P\cap Q$ = QR$Q\cap R$ = RP$R\cap P$ = ϕ$\phi$

But, PQRS$P\cap Q\cap R ≠ S$

Question 6

An experiment consists of throwing up of a dice and includes events A,B and C:

A = an even number on throwing of the first die

B = an odd number on throwing of the first die

C = sum of numbers on the dice will less than equals to 5.

(a) A     (b) not B       (c) A or B     (d) A and B

(e) A but not C     (f) B or C     (g) B or C    (h) ABC$A\cap B’\cap C’$

Sol:

Sample space when a pair of dice is rolled (S) = { (x,y) : x,y = 6,5,4,32,1 }

S = { (6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6)

(5,1) , (5,2)  , (5,3) , (5,4) , (5,5) , (5,6)

(4,1) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6)

(3,1) , (3,2)  , (3,3) , (3,4) , (3,5) , (3,6)

(2,1) , (2,2)  , (2,3) , (2,4) , (2,5) , (2,6)

(1,1) , (1,2)  , (1,3) , (1,4) , (1,5) , (1,6)}

A = { (6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6) ,    (4,1) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6), (2,1), (2,2) , (2,3) , (2,4) , (2,5) , (2,6) }

B = { (5,1) , (5,2)  , (5,3) , (5,4) , (5,5) , (5,6) , (3,1) , (3,2)  , (3,3) , (3,4) , (3,5) , (3,6), (1,1) ,(1,2)  , (1,3) , (1,4) , (1,5) , (1,6) }

C = { (1,1) , (1,2)  , (1,3) , (1,4), (2,1) , (2,2)  , (2,3),(3,1) , (3,2)  , (4,1)}

(a) A’ = {(5,1) , (5,2)  , (5,3) , (5,4) , (5,5) , (5,6), (3,1) , (3,2)  , (3,3) , (3,4) , (3,5) , (3,6), (1,1), (1,2)  , (1,3) , (1,4) , (1,5) , (1,6) } = B

(b) Not B = B’ = { (6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6) , (4,1) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6) , (2,1) , (2,2)  , (2,3) , (2,4) , (2,5) , (2,6) } = A

(c) A or B = AB$A\cup B$ = { (6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6)

(5,1) , (5,2)  , (5,3) , (5,4) , (5,5) , (5,6)

(4,1) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6)

(3,1) , (3,2)  , (3,3) , (3,4) , (3,5) , (3,6)

(2,1) , (2,2)  , (2,3) , (2,4) , (2,5) , (2,6)

(1,1) , (1,2)  , (1,3) , (1,4) , (1,5) , (1,6)}

(d) A and B = AB$A\cap B$ = ϕ$\phi$

(e) A but not C = A – C = {(6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6) , (2,4) , (2,5) , (2,6) }

(f) B or C = BC$B\cup C$ = {(5,1) , (5,2)  , (5,3) , (5,4) , (5,5) , (5,6) , (4,1) , (3,1) , (3,2)  , (3,3) , (3,4) , (3,5) , (3,6) , (2,1) , (2,2)  , (2,3),  (1,1) , (1,2)  , (1,3) , (1,4) , (1,5) , (1,6) }

(g) B and C = AB$A\cap B$ = { (1,1) , (1,2)  , (1,3) , (1,4) , (3,1) , (3,2)  }

(h) C’ = {(6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6), (5,1) , (5,2)  , (5,3) , (5,4) , (5,5) , (5,6) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6) , (3,3) , (3,4) , (3,5) , (3,6) , (2,5) , (2,6) , (1,5) , (1,6) }

Therefore, ABC$A\cap B’\cap C’$ = AAC$A\cap A\cap C’$ = AC$A\cap C’$

= {(6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6) , (2,4) , (2,5) , (2,6)}

Question 7

An experiment consists of throwing up of a dice and includes A, B and C.

A = an even number on throwing of the first die

B = an odd number on throwing of the first die

C = sum of numbers on the dice will less than equals to 5.

Give answer in true or false.

(a) B and A are mutually exclusive events.

(b) B and A are mutually exclusive and exhaustive events.

(c) A = B’

(d) A and C are mutually exclusive and exhaustive events.

(e) B’ and A are mutually exclusive events.

(f) A’ , B’ & C are mutually exhaustive and exclusive.

Sol:

A = {(6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6) , (2,1) , (2,2)  , (2,3) , (2,4) , (2,5) , (2,6) , (4,1) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6)}

B= {(5,1) , (5,2)  , (5,3) , (5,4) , (5,5) , (5,6) , (3,1) , (3,2)  , (3,3) , (3,4) , (3,5) , (3,6) , (1,1) , (1,2)  , (1,3) , (1,4) , (1,5) , (1,6)}

C = { (4,1) , (3,1) , (3,2)  , (2,1) , (2,2)  , (2,3) , (1,1) , (1,2)  , (1,3) , (1,4)}

(a) AB=ϕ$A\cap B = \phi$

B & A are mutually exclusive events.

The statement is true.

(b) AB=ϕ$A\cap B = \phi$ and AB=S$A\cup B = S$

B & A are mutually exclusive and exhaustive events.

The statement is true.

(c) B’ = {(4,1) , (4,2)  , (4,3) , (4,4) , (4,5) , (4,6) , (6,1) , (6,2)  , (6,3) , (6,4) , (6,5) , (6,6) , (2,1) , (2,2)  , (2,3) , (2,4) , (2,5) , (2,6)} = A

The statement is true.

(d) AC$A\cap C$ = {(2,1) , (2,2)  , (2,3), (4,1)} = ϕ$\phi$

C and A are not mutually exclusive events.

The statement is false.

(e) AB=AA=A$A\cap B’ = A\cap A = A$

AB$A\cap B$ = ϕ$\phi$

B’ and A are not mutually exclusive events.

The statement is false.

(f) ABC$A’\cap B’\cap C$ = S

BC$B\cap C$ = { (4,1) , (2,3) , (2,2) , (2,1) } =  ϕ$\phi$

Events B’, C and A’ are not mutually exhaustive and exclusive.

The statement is false.