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 ≤ 5
Describe the events  (i) A'  (ii) not B  (iii) A or B  (iv) A and B
(v) A but not C  (vi) B or C  (vii) B and C  (viii) A ∩ B' ∩ C'

Solution:

When two dice are thrown, the sample space is given by

S = {(x, y) : x, y ∈ {1, 2, 3, 4, 5, 6} }

Hence,

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

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

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

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

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

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

Accordingly,

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

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

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

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

B = (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)}

(i)

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

A' = S - A = (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)         = B

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

(ii)

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

not B = B' = S - B = (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) = A

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

(iii)

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

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

A or B = (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)   = S

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

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

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

(iv) A and B = A ∩ B = Φ

(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)}

(vi)

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

B or C = B υ C = (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)}

(vii) B and C = B ∩ C = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)}

(viii) First we will find C'.

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

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

C' = (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1),

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

Therefore,

A ∩ B' ∩ C' = A ∩ A ∩ 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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NCERT Solutions Class 11 Maths Chapter 16 Exercise 16.2 Question 6

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 ≤ 5. Describe the events (i) A' (ii) not B (iii) A or B (iv) A and B (v) A but not C (vi) B or C (vii) B and C (viii) A ∩ B' ∩ C'

Summary:

(i) 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)} (ii) not 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)} (iii) A or B = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} (iv) A and B = A ∩ B = Φ (v) A but not 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)} (vi) B or 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)} (vii) B and C = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)} (viii) A ∩ B' ∩ 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)}