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#16.2

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

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

Here,

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

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

Hence A' and B' are mutually exclusive.

Again,

$\therefore$ B' $\cap$ C = A $\cap$ C $\neq$ $\phi$ (from (iv))

Hence B' and C are not mutually exclusive.

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

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#16.2

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

(v) $A$ 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))

$\therefore$ A $\cap$ B’ $\neq \phi$

Hence A and B' not mutually exclusive. FALSE

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#16.2

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

(iv) A and C are mutually exclusive

Here,

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

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#16.2

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

(iii) $A=B{}'$

Here,

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

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#16.2

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

(ii) 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

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#16.2

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

(i) 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

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#16.2

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

(viii) $A\cap {B}'\cap {C}'$

Sample space when two dice are thrown:

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

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

The possible sum are 2,3,4,5

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

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#16.2

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

              (vii) B and C

Sample space when two dice are thrown:

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

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

The possible sum are 2,3,4,5

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

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#16.2

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

(vi) B or C

Sample space when two dice are thrown:

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

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

The possible sum are 2,3,4,5

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

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#16.2

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   $\leq 5$

Describe the events

(v) A but not C

Sample space when two dice are thrown:

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

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

The possible sum are 2,3,4,5

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

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7.(vi) Refer to question 6 above, state true or false: (give reason for your answer) (vi) are mutually exclusive and exhaustive.

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

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7.(iv) Refer to question 6 above, state true or false: (give reason for your answer) (iv) A and C are mutually exclusive

Posted by

HARSH KANKARIA

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

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JEE Main high-scoring chapters and topics

7.(ii) Refer to question 6 above, state true or false: (give reason for your answer) (ii) A and B are mutually exclusive and exhaustive

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

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

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

(v) $A$ and ${B}'$ are mutually exclusive.

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

(iv) A and C are mutually exclusive

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

(iii) $A=B{}'$

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

(ii) A and B are mutually exclusive and exhaustive

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

(i) A and B are mutually exclusive

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

(viii) $A\cap {B}'\cap {C}'$

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

(vii) B and C

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

(vi) B or C

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 $\leq 5$

Describe the events

(v) A but not C