NCERT Solutions for Class 11 Maths Chapter 14 Exercise 14.1 - Probability

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?

Answer:

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 $\cap$ F = {4} $\cap$ {2, 4, 6}

= {4} $\ne \varphi$

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

Answer:

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

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ 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

Answer:

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

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

Given, B: a number greater than 7

As no number on the die is greater than 7

B = $\pi 100\varphi$

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

C: a multiple of 3.

Answer:

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

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ 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

Answer:

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

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ 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

Answer:

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

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ 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

Answer:

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

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

Given, F : a number not less than 3

F = {x: x $\in$ S, x $\ge$ 3 } = {3, 4, 5, 6}

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

Also find (a) $A\cup B$

Answer:

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

B= $\varphi$

$\therefore$ A $\cup$ B = {1, 2, 3, 4, 5, 6} $\cup$$\varphi$ = {1, 2, 3, 4, 5, 6}

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

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

Answer:

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

B= $\varphi$

$\therefore$ A $\pi 80\cap$ B = {1, 2, 3, 4, 5, 6} $\pi 80\cap$$\varphi$ = $\varphi$

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

Also find (c) $B\cup C$

Answer:

B= $\varphi$

C= {3, 6}

$\therefore$ B $\cup$ C = $\varphi$$\cup$ {3, 6} = {3, 6}

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

(d) Also find $E\cap F$

Answer:

E = {6}

F = {3, 4, 5, 6}

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

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

Also find (e) $D\cap E$

Answer:

D = {1, 2, 3}

E = {6}

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

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

Also find (f) $A-C$

Answer:

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

C = {3, 6}

$\therefore$ 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) $D-E$

Answer:

D = {1, 2, 3}

E = {6}

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

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

Also find (h) $E\cap F^{\prime }$

Answer:

E = {6}

F = {3, 4, 5, 6}

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

$\therefore$ E $\pi 80\cap$ F' = {6} $\pi 80\cap$ {1, 2} = $\varphi$

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

Also find (i) $F^{\prime }$

Answer:

F = {3, 4, 5, 6}

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

Question:3(a) An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

the sum is greater than $8$

Answer:

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

Question:3(b) An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

$2$ occurs on either die

Answer:

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 $\pi 100\in$ 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) $\pi 100\in$ 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)}

Question:3(c). An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

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

Answer:

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

Question:3(d). An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

Which pairs of these events are mutually exclusive?

Answer:

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 = $\varphi$ (no common element in A and B)

Hence, A and B are mutually exclusive

Again, B $\cap$ C = $\varphi$ (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.

Question:4(i) Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event” three tails show and D denote the event ‘a head shows on the first coin”. Which events are

mutually exclusive?

Answer:

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 = $\varphi$

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

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

C $\cap$ D = {TTT} $\cap$ {HHH, HHT, HTH, HTT} = $\varphi$ ; 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} = $\varphi$ ; 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.

Question:4.(ii) Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event” three tails show and D denote the event ‘a head shows on the first coin”. Which events are

simple?

Answer:

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.

Question:4.(iii) Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event” three tails show and D denote the event ‘a head shows on the first coin”. Which events are

Compound?

Answer:

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.

Answer:

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} = $\varphi$ ; Hence A and B are mutually exclusive.

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

Three events which are mutually exclusive and exhaustive.

Answer:

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 = $\varphi$ ; B $\cap$ C = $\varphi$ ; C $\cap$ A = $\varphi$

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.

Answer:

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} $\ne$ $\varphi$

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.

Answer:

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 = $\varphi$

Hence, A and B are mutually exclusive.

Also, A $\cup$ B $\ne$ 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

Answer:

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 = $\varphi$ ; B $\cap$ C = $\varphi$ ; C $\cap$ A = $\varphi$

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} $\ne$ S

Hence A, B and C are not exhaustive events.

Question:6.(i) 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 $\le 5$.

Describe the events

$A{}^{\prime }$

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\le$ x,y $\le$ 6}

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

Describe the events

not B

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\le$ x,y $\le$ 6}

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

Describe the events

A or B

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\le$ x,y $\le$ 6}

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

Describe the events

A and B

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\le$ x,y $\le$ 6}

A: getting an even number on the first die = {(a,b): a $\in$ {2,4,6} and 1 $\le$ b $\le$ 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 $\in$ {1,3,5} and 1 $\le$ b $\le$ 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' = $\varphi$ (From (ii))

Question:6.(v) 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 $\le 5$

Describe the events

A but not C

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\le$ x,y $\le$ 6}

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

The possible sum are 2,3,4,5

C = {(a,b): 2 $\le$ a + b $\le$ 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 $\le 5$

Describe the events

B or C

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\le$ x,y $\le$ 6}

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

The possible sum are 2,3,4,5

C = {(a,b): 2 $\le$ a + b $\le$ 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 $\le 5$

Describe the events

B and C

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\le$ x,y $\le$ 6}

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

The possible sum are 2,3,4,5

C = {(a,b): 2 $\le$ a + b $\le$ 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 $\le 5$

Describe the events

$A\cap B^{\prime }\cap C^{\prime }$

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\le$ x,y $\le$ 6}

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

The possible sum are 2,3,4,5

C = {(a,b): 2 $\le$ a + b $\le$ 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)}

Question:7.(i) Refer to question 6 above, state true or false: (give reason for your answer)

A and B are mutually exclusive

Answer:

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 = $\varphi$

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

Hence, A and B are mutually exclusive. TRUE

Question:7.(ii) Refer to question 6 above, state true or false: (give reason for your answer)

A and B are mutually exclusive and exhaustive

Answer:

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 = $\varphi$

A $\cap$ B = $\varphi$ , 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

Question:7.(iii) Refer to question 6 above, state true or false: (give reason for your answer)

$A=B{}^{\prime }$

Answer:

Here,

S = {(x,y): 1 $\le$ x,y $\le$ 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

Question:7.(iv) Refer to question 6 above, state true or false: (give reason for your answer)

A and C are mutually exclusive

Answer:

Here,

S = {(x,y): 1 $\le$ x,y $\le$ 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 = $\varphi$

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

Hence, A and B are not mutually exclusive. FALSE

Question:7.(v) Refer to question 6 above, state true or false: (give reason for your answer)

$A$ and $B^{\prime }$ are mutually exclusive.

Answer:

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

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

$\therefore$ A $\cap$ B’ $\ne \varphi$

Hence A and B' not mutually exclusive. FALSE

Question:7.(vi) Refer to question 6 above, state true or false: (give reason for your answer)

$A^{\prime },B^{\prime },C$ are mutually exclusive and exhaustive.

Answer:

Here,

S = {(x,y): 1 $\le$ x,y $\le$ 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 = $\varphi$

$\therefore$ A' $\cap$ B' = B $\cap$ A = $\varphi$ (from (iii) and (i))

Hence A' and B' are mutually exclusive.

Again,

$\therefore$ B' $\cap$ C = A $\cap$ C $\ne$ $\varphi$ (from (iv))

Hence B' and C are not mutually exclusive.

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