Q5. Two dice are thrown. List the event C: “the sum is at least 7 and a multiple of 3” (NCERT Exercise 14.1, Q3).
The sum must be 9 or 12: C = {(3, 6), (4, 5), (5, 4), (6, 3), (6, 6)}.
Q6. Two dice are thrown. A: an even number on the first die; B: an odd number on the first die; C: the sum is at most 5. Describe “B and C” and count “A but not C” (NCERT Exercise 14.1, Q6).
B ∩ C = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)}. A − C holds the outcomes with an even first die and a sum of at least 6: 3 with a first 2, 5 with a first 4 and 6 with a first 6, so 14 outcomes.
Q7. For the events of Q6, say true or false, with reasons: (i) A and B are mutually exclusive and exhaustive; (ii) A = B′; (iii) A and B′ are mutually exclusive (NCERT Exercise 14.1, Q7 (ii), (iii) and (v)).
(i) True: the first die is either even or odd, never both. (ii) True: not odd means even. (iii) False: B′ = A, and A ∩ A = A ≠ ϕ.
Q8. Three coins are tossed. Give two events that are mutually exclusive but not exhaustive, and three such events (NCERT Exercise 14.1, Q5).
For example, “the first coin shows a head” {HHH, HHT, HTH, HTT} and “the first two coins show tails” {TTH, TTT}, which leave out THH and THT; and “exactly one head”, “exactly two heads” and “three heads”, which leave out TTT.
Q9. For one roll of a die, A = “a prime number” and B = “an odd number”. Find “exactly one of A and B” and “neither A nor B”.
Exactly one: (A − B) ∪ (B − A) = {2} ∪ {1} = {1, 2}. Neither: (A ∪ B)′ = {4, 6}.
Q10. Show that for any events A and B, the events A ∩ B and A ∩ B′ are mutually exclusive and that their union is A.
An outcome cannot be both in B and in B′, so (A ∩ B) ∩ (A ∩ B′) = ϕ. Every outcome of A is either in B or not, so (A ∩ B) ∪ (A ∩ B′) = A.
Q11. Taking S as the 8 ordered outcomes of three tosses, how many events, and how many simple events, are there?
S has 8 outcomes, so there are 28 = 256 events, of which 8 are simple.
Q12. A coin is tossed three times. Are “at least two heads” and “at least two tails” mutually exclusive and exhaustive?
Yes. With three tosses, the heads and tails add up to 3, so at least two heads means at most one tail. Exactly one of the two events occurs every time.