**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

=

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, **

**Sol:**

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

(b) Q =

(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:

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:

(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.

**Q****uestion 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:

P = no heads

Q = no tails

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

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

P = no heads

Q = exactly 1 head

R = at least 2 heads

P = {TTT}

Q = {TTH, THT, HTT}

R = {THH, HTH, HHT, HHH}

Because

(c) 2 events are not mutually exclusive:

P = 3 heads

Q = at least 2 heads

P = {HHH}

Q = {THH , HTH , HHT , HHH }

Because

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

P = exactly 1 head

Q = exactly 1 tail

P = { TTH , THT , HTT }

Q = {THH , HTH ,HHT }

Because

But,

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

P = exactly 3 heads

Q = 1 head and 2 tails

R = 1 tail and 2 heads

P = { HHH }

Q = { TTH , THT , HTT }

R = { THH , HTH , HHT }

Because

But,

**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) A∩B′∩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 =

(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 =

(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 =

(g) B and C =

(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,

= {(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)

B & A are mutually exclusive events.

The statement is true.

(b)

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)

C and A are not mutually exclusive events.

The statement is false.

(e)

B’ and A are not mutually exclusive events.

The statement is false.

(f)

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

The statement is false.