NCERT Solutions for Exercise 16.2 Class 11 Maths Chapter 16 - Probability

# NCERT Solutions for Exercise 16.2 Class 11 Maths Chapter 16 - Probability

Edited By Ravindra Pindel | Updated on Jul 12, 2022 04:41 PM IST

In the previous exercise, you have learned about random experiments and their outcomes, sample space. In the NCERT solutions for Class 11 Maths chapter 16 exercise 16.2, you will learn about the occurrence of an event, types of events, and algebra of events. There are mainly four types of events called an impossible event, sure events, simple events, and compound events in the Class 11 Maths chapter 16 exercise 16.2. Algebra of an event, exclusive events, and exhaustive events are other topics that are very important in Class 11 Maths chapter 16 exercise 16.2. Two events are called mutually exclusive events if they can't occur simultaneously or if one event has occurred then the second event won't occur. If an experiment is performed and at least one of n events necessarily occurs then such an event is called an exhaustive event.

In Class 11 Maths chapter 16 exercise 16.2 solutions, you will get questions related to exclusive and exhaustive events also. In this NCERT book exercise, you will also learn about the disjoint set. If you are looking for NCERT Solutions, click on the given link to get NCERT solutions from Class 6 to Class 12 for Science and Math at one place.

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## Question:1. A die is rolled. Let E be the event “die shows $4$” and F be the event “die shows even number”. Are E and F mutually exclusive?

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6}

Now,

E = event that the die shows 4 = {4}

F = event that the die shows even number = {2, 4, 6}

E $\dpi{100} \cap$ F = {4} $\dpi{100} \cap$ {2, 4, 6}

= {4} $\dpi{100} \neq \phi$

Hence E and F are not mutually exclusive event.

Question:2(i) A die is thrown. Describe the following events:

A: a number less than 7

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\dpi{100} \in$ N, x<7}

Given, A : a number less than 7

As every number on a die is less than 7

A = {1, 2, 3, 4, 5, 6} = S

Question:2(ii) A die is thrown. Describe the following events:

B: a number greater than 7

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\dpi{100} \in$ N, x<7}

Given, B: a number greater than 7

As no number on the die is greater than 7

B = $\dpi{100} \phi$

Question:2(iii) A die is thrown. Describe the following events:

C: a multiple of 3.

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\dpi{100} \in$ N, x<7}

Given, C : a multiple of 3

C = {3, 6}

Question:2(iv) A die is thrown. Describe the following events:

D: a number less than 4

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\dpi{100} \in$ N, x<7}

Given, D : a number less than 4

D = {1, 2, 3}

Question:2(v) A die is thrown. Describe the following events:

E: an even multiple greater than 4

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\dpi{100} \in$ N, x<7}

Given, E : an even number greater than 4

S1 = Subset of S containing even numbers = {2,4,6}

Therefore , E = {6}

Question:2(vi). A die is thrown. Describe the following events:

F: a number not less than 3

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\dpi{100} \in$ N, x<7}

Given, F : a number not less than 3

F = {x: x $\dpi{100} \in$ S, x $\dpi{80} \geq$ 3 } = {3, 4, 5, 6}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (a) $A\cup B$

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

B= $\dpi{100} \phi$

$\dpi{100} \therefore$ A $\dpi{80} \cup$ B = {1, 2, 3, 4, 5, 6} $\dpi{80} \cup$$\dpi{100} \phi$ = {1, 2, 3, 4, 5, 6}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (b) $A\cap B$.

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

B= $\dpi{100} \phi$

$\dpi{100} \therefore$ A $\dpi{80} \cap$ B = {1, 2, 3, 4, 5, 6} $\dpi{80} \cap$$\dpi{100} \phi$ = $\dpi{100} \phi$

Question:2.(vi) A die is thrown. Describe the following events:

Also find (c) $B\cup C$

B= $\dpi{100} \phi$

C= {3, 6}

$\dpi{100} \therefore$ B $\dpi{80} \cup$ C = $\dpi{100} \phi$$\dpi{80} \cup$ {3, 6} = {3, 6}

Question:2.(vi) A die is thrown. Describe the following events:

(d) Also find $E\cap F$

E = {6}

F = {3, 4, 5, 6}

$\dpi{100} \therefore$ E $\dpi{80} \cap$ F = {6} $\dpi{80} \cap$ {3, 4, 5, 6} = {6}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (e) $D\cap E$

D = {1, 2, 3}

E = {6}

$\dpi{100} \therefore$ D $\dpi{80} \cap$ E = {1, 2, 3} $\dpi{80} \cap$ {6} = $\dpi{100} \phi$ (As nothing is common in these sets)

Question:2.(vi) A die is thrown. Describe the following events:

Also find (f) $A-C$

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

C = {3, 6}

$\dpi{100} \therefore$ A - C = {1, 2, 3, 4, 5, 6} - {3, 6} = {1, 2, 4, 5}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (g) $D-E$

D = {1, 2, 3}

E = {6}

$\dpi{100} \therefore$ D - E = {1, 2, 3} - {6} = {1, 2, 3}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (h) $E\cap F'$

E = {6}

F = {3, 4, 5, 6}

$\dpi{100} \therefore$ F' = {3, 4, 5, 6}' = S - F = {1, 2}

$\dpi{100} \therefore$ E $\dpi{80} \cap$ F' = {6} $\dpi{80} \cap$ {1, 2} = $\dpi{100} \phi$

Question:2.(vi) A die is thrown. Describe the following events:

Also find (i) ${F}'$

F = {3, 4, 5, 6}

$\dpi{100} \therefore$ F' = {3, 4, 5, 6}' = S - F = {1, 2}

the sum is greater than $8$

Sample space when a die is rolled:

S = {1, 2, 3, 4, 5, 6}

Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes]

E = [ {(x,y): x,y $\dpi{100} \in$ S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)}

Now,

A : the sum is greater than 8

Possible sum greater than 8 are 9, 10, 11 and 12

A = [ {(a,b): (a,b) $\dpi{100} \in$ E, a+b>8 } ]= {(3,6), (4,5), (5, 4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)}

$2$ occurs on either die

Sample space when a die is rolled:

S = {1, 2, 3, 4, 5, 6}

Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes]

E = [ {(x,y): x,y $\dpi{100} \in$ S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)}

Now,

B: 2 occurs on either die

Hence the number 2 can come on first die or second die or on both the die simultaneously.

B = [ {(a,b): (a,b) $\dpi{100} \in$ E, a or b = 2 } ]= {(1,2), (2,2), (3, 2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)}

the sum is at least $7$ and a multiple of $3$

Sample space when a die is rolled:

S = {1, 2, 3, 4, 5, 6}

Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes]

E = [ {(x,y): x,y $\dpi{100} \in$ S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)}

Now,

C: the sum is at least 7 and a multiple of 3

The sum can only be 9 or 12.

C = [ {(a,b): (a,b) $\dpi{100} \in$ E, a+b>6 & a+b = 3k, k $\dpi{100} \in$ I} ]= {(3,6), (6,3), (5, 4), (4,5), (6,6)}

Which pairs of these events are mutually exclusive?

For two elements to be mutually exclusive, there should not be any common element amongst them.

Also, A = {(3,6), (4,5), (5, 4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)}

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

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

Now, A $\cap$ B = $\phi$ (no common element in A and B)

Hence, A and B are mutually exclusive

Again, B $\cap$ C = $\phi$ (no common element in B and C)

Hence, B and C are mutually exclusive

Again, C $\cap$ A = {(3,6), (6,3), (5, 4), (4,5), (6,6)}

Therefore,

A and B, B and C are mutually exclusive.

mutually exclusive?

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Now,

A = Event that three heads show up = {HHH}

B = Event that two heads and one tail show up = {HHT, HTH, THH}

C = Event that three tails show up = {TTT}

D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}

(i). For two elements X and Y to be mutually exclusive, X $\cap$ Y = $\phi$

A $\cap$ B = {HHH} $\cap$ {HHT, HTH, THH} = $\phi$ ; Hence A and B are mutually exclusive.

B $\cap$ C = {HHT, HTH, THH} $\cap$ {TTT} = $\phi$ ; Hence B and C are mutually exclusive.

C $\cap$ D = {TTT} $\cap$ {HHH, HHT, HTH, HTT} = $\phi$ ; Hence C and D are mutually exclusive.

D $\cap$ A = {HHH, HHT, HTH, HTT} $\cap$ {HHH} = {HHH} ; Hence D and A are not mutually exclusive.

A $\cap$ C = {HHH} $\cap$ {TTT} = $\phi$ ; Hence A and C are mutually exclusive.

B $\cap$ D = {HHT, HTH, THH} $\cap$ {HHH, HHT, HTH, HTT} = {HHT, HTH} ; Hence B and D are not mutually exclusive.

simple?

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Now,

A = Event that three heads show up = {HHH}

B = Event that two heads and one tail show up = {HHT, HTH, THH}

C = Event that three tails show up = {TTT}

D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}

(ii).If an event X has only one sample point of a sample space, it is called a simple event.

A = {HHH} and C = {TTT}

Hence, A and C are simple events.

Compound?

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Now,

A = Event that three heads show up = {HHH}

B = Event that two heads and one tail show up = {HHT, HTH, THH}

C = Event that three tails show up = {TTT}

D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}

(iv). If an event has more than one sample point, it is called a Compound event.

B = {HHT, HTH, THH} and D = {HHH, HHT, HTH, HTT}

Hence, B and D are compound events.

Question:5(i) Three coins are tossed. Describe

Two events which are mutually exclusive.

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

(i)

A = Event that three heads show up = {HHH}

B = Event that three tails show up = {TTT}

A $\cap$ B = {HHH} $\cap$ {TTT} = $\phi$ ; Hence A and B are mutually exclusive.

Question:5(ii) Three coins are tossed. Describe

Three events which are mutually exclusive and exhaustive.

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Let ,

A = Getting no tails = {HHH}

B = Getting exactly one tail = {HHT, HTH, THH}

C = Getting at least two tails = {HTT, THT, TTH}

Clearly, A $\cap$ B = $\phi$ ; B $\cap$ C = $\phi$ ; C $\cap$ A = $\phi$

Since (A and B), (B and C) and (A and C) are mutually exclusive

Therefore A, B and C are mutually exclusive.

Also,

A $\cup$ B $\cup$ C = S

Hence A, B and C are exhaustive events.

Hence, A, B and C are three events which are mutually exclusive and exhaustive.

Question:5(iii). Three coins are tossed. Describe

Two events, which are not mutually exclusive.

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Let ,

A = Getting at least one head = {HHH, HHT, HTH, THH, TTH}

B = Getting at most one head = {TTH, TTT}

Clearly, A $\cap$ B = {TTH} $\neq$ $\phi$

Hence, A and B are two events which are not mutually exclusive.

Question:5.(iv) Three coins are tossed. Describe

Two events which are mutually exclusive but not exhaustive.

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Let ,

A = Getting exactly one head = {HTT, THT, TTH}

B = Getting exactly one tail = {HHT, HTH, THH}

Clearly, A $\cap$ B = $\phi$

Hence, A and B are mutually exclusive.

Also, A $\cup$ B $\neq$ S

Hence, A and B are not exhaustive.

Question:5.(v) Three coins are tossed. Describe

Three events which are mutually exclusive but not exhaustive

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Let ,

A = Getting exactly one tail = {HHT, HTH, THH}

B = Getting exactly two tails = {HTT, TTH, THT}

C = Getting exactly three tails = {TTT}

Clearly, A $\cap$ B = $\phi$ ; B $\cap$ C = $\phi$ ; C $\cap$ A = $\phi$

Since (A and B), (B and C) and (A and C) are mutually exclusive

Therefore A, B and C are mutually exclusive.

Also,

A $\cup$ B $\cup$ C = {HHT, HTH, THH, HTT, TTH, THT, TTT} $\neq$ S

Hence A, B and C are not exhaustive events.

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice $\leq 5$.

Describe the events

$A{}'$

Sample space when two dice are thrown:

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

A: getting an even number on the first die = {(a,b): a $\dpi{80} \in$ {2,4,6} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 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), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

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

= B : getting an odd number on the first die.

Question:6.(ii) Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice $\leq 5$.

Describe the events

not B

Sample space when two dice are thrown:

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

B: getting an odd number on the first die = {(a,b): a $\dpi{80} \in$ {1,3,5} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 6}

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

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

= A : getting an even number on the first die.

Question:6.(iii) Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice $\leq 5$.

Describe the events

A or B

Sample space when two dice are thrown:

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

A: getting an even number on the first die = {(a,b): a $\dpi{80} \in$ {2,4,6} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 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), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B: getting an odd number on the first die = {(a,b): a $\dpi{80} \in$ {1,3,5} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 6}

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

(iii) A or B = A $\cup$ B = {(1,1), (1,2) .... (1,6), (3,1), (3,2).... (3,6), (5,1), (5,2)..... (5,6), (2,1), (2,2)..... (2,6), (4,1), (4,2)..... (4,6), (6,1), (6,2)..... (6,6)} = S

Question:6.(iv) Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice $\leq 5$

Describe the events

A and B

Sample space when two dice are thrown:

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

A: getting an even number on the first die = {(a,b): a $\dpi{80} \in$ {2,4,6} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 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), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B: getting an odd number on the first die = {(a,b): a $\dpi{80} \in$ {1,3,5} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 6}

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

(iii) A and B = A $\cap$ B = A $\cap$ A' = $\phi$ (From (ii))

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice $\leq 5$

Describe the events

A but not C

Sample space when two dice are thrown:

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

A: getting an even number on the first die = {(a,b): a $\dpi{80} \in$ {2,4,6} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 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), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

C: getting the sum of the numbers on the dice $\dpi{80} \leq$ 5

The possible sum are 2,3,4,5

C = {(a,b): 2 $\dpi{80} \leq$ a + b $\dpi{80} \leq$ 5} = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

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

Question:6.(vi) Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice $\leq 5$

Describe the events

B or C

Sample space when two dice are thrown:

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

B: getting an odd number on the first die = {(a,b): a $\dpi{80} \in$ {1,3,5} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 6}

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

C: getting the sum of the numbers on the dice $\dpi{80} \leq$ 5

The possible sum are 2,3,4,5

C = {(a,b): 2 $\dpi{80} \leq$ a + b $\dpi{80} \leq$ 5} = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

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

Question:6.(vii) Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice $\leq 5$

Describe the events

B and C

Sample space when two dice are thrown:

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

B: getting an odd number on the first die = {(a,b): a $\dpi{80} \in$ {1,3,5} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 6}

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

C: getting the sum of the numbers on the dice $\dpi{80} \leq$ 5

The possible sum are 2,3,4,5

C = {(a,b): 2 $\dpi{80} \leq$ a + b $\dpi{80} \leq$ 5} = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

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

Question:6.(viii) Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice $\leq 5$

Describe the events

$A\cap {B}'\cap {C}'$

Sample space when two dice are thrown:

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

A: getting an even number on the first die = {(a,b): a $\dpi{80} \in$ {2,4,6} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 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), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B: getting an odd number on the first die = {(a,b): a $\dpi{80} \in$ {1,3,5} and 1 $\dpi{80} \leq$ b $\dpi{80} \leq$ 6}

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

C: getting the sum of the numbers on the dice $\dpi{80} \leq$ 5

The possible sum are 2,3,4,5

C = {(a,b): 2 $\dpi{80} \leq$ a + b $\dpi{80} \leq$ 5} = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(viii) A $\cap$ B' $\cap$ C' = A $\cap$ A $\cap$ C' (from (ii))

= A $\cap$ C' = A - C = {(2,4), (2,5), (2,6), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

A and B are mutually exclusive

Here,

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

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

(i) X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

A $\cap$ B = $\phi$ , since A and B have no common element amongst them.

Hence, A and B are mutually exclusive. TRUE

A and B are mutually exclusive and exhaustive

Here,

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

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

(ii) X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

A $\cap$ B = $\phi$ , since A and B have no common element amongst them.

Hence, A and B are mutually exclusive.

Also,

A $\cup$ B = {(2,1), (2,2).... (2,6), (4,1), (4,2).....(4,6), (6,1), (6,2)..... (6,6), (1,1), (1,2).... (1,6), (3,1), (3,2)..... (3,6), (5,1), (5,2).... (5,6)} = S

Hence, A and B are exhaustive.

TRUE

$A=B{}'$

Here,

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

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

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

(iii) Therefore, B' = S -B = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} = A

TRUE

A and C are mutually exclusive

Here,

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

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

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

(iv) X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

A $\cap$ C = {(2,1), (2,2), (2,3), (4,1)} ,

Hence, A and B are not mutually exclusive. FALSE

$A$ and ${B}'$ are mutually exclusive.

X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

A $\cap$ B' = A $\cap$ A = A (From (iii))

$\dpi{100} \therefore$ A $\cap$ B’ $\dpi{100} \neq \phi$

Hence A and B' not mutually exclusive. FALSE

${A}',{B}',C$ are mutually exclusive and exhaustive.

Here,

S = {(x,y): 1 $\dpi{80} \leq$ x,y $\dpi{80} \leq$ 6}

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

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

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

(vi) X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

$\dpi{100} \therefore$ A' $\cap$ B' = B $\cap$ A = $\phi$ (from (iii) and (i))

Hence A' and B' are mutually exclusive.

Again,

$\dpi{100} \therefore$ B' $\cap$ C = A $\cap$ C $\dpi{80} \neq$ $\phi$ (from (iv))

Hence B' and C are not mutually exclusive.

Hence, A', B' and C are not mutually exclusive and exhaustive. FALSE

## More About NCERT Solutions for Class 11 Maths Chapter 16 Exercise 16.2:-

In Class 11 Maths chapter 16 exercise 16.2 solutions, you will get questions related to exhaustive, exclusive events and algebra of events There are seven questions in Class 11 Maths chapter 16 exercise 16.2, which you can solve to get conceptual clarity. You are advised to go through the solved examples given before this exercise which will help you to solve the exercise problems very easily.

Also Read| Probability Class 11 Notes

## Benefits of NCERT Solutions for Class 11 Maths Chapter 16 Exercise 16.2:-

• Exercise 16.2 Class 11 Maths is a very important exercise in this chapter as it is consists of questions related to an event of a random experiment.

• Many times question from Class 11 Maths chapter 16 exercise 16.2 is directly asked in the final exams.
• The concepts from Class 11 Maths chapter 16 exercise 16.2 are very useful for the this exerices as well as in upcoming classes.

Also see-

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## Subject Wise NCERT Exampler Solutions

Happy learning!!!

1. A die is rolled then what is the probability of getting number 6 on the die face ?

The probability of getting number 6 on the die face is = 1/6

2. A die is rolled then what is the probability of getting number 7 on the die face ?

The probability of getting the number '7' on the die face is zero.

3. When a biased coin is tossed and the probability of getting head on the coin is 0.61, then what is the probability of getting tail ?

The probability of getting head = 0.61

Probability of getting tail = 1- 0.61 = 0.39

4. What is the probability of a sure event ?

The probability of a sure event is 1.

5. what is the probability of an impossible event ?

The probability of an impossible event is zero.

6. If the probability of an event A is 0.64 then what is the probability of its compliment event ?

The probability of compliment of A p(A') = 1-p(A) = 0.36.

7. If the probability of an event A is zero the does A is an impossible event ?

Yes, If the probability of an event A is zero then A is an impossible event.

8. What is equal likely outcomes ?

If the probability of all outcomes of a random experiment is the same then such outcomes are called equal likely outcomes.

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