(i) Probability that both of them are selected
P (both of them are selected) = P (A ∩ B) = P (A) × P (B)
Substituting the values
= 1/6 × 1/4
= 1/24
Hence, the probability that both of them are selected is 1/24.
(ii) Probability that only one of them is selected
P (only one of them is selected) = P (A and not B or B and not A)
= P (A and not B) + P (B and not A)
We get
Substituting the values
= (1/6 × 3/4) + (1/4 × 5/6)
= 3/24 + 5/24
So we get
= 1/3
Hence, the probability that only one of them is selected is 1/3.
(iii) Probability that none is selected
P (none is selected)
Substituting the values
= 5/6 × 3/4
= 5/8
Hence, the probability that none is selected is 5/8.
(iv) Probability that at least one of them is selected
P (at least one of them is selected) = P (selecting only A) + P (selecting only B) + P (selecting both)
We know that
= P (A and not B) + P (B and not A) + P (A and B)
It can be written as
Substituting the values
= (1/6 × 3/4) + (1/4 × 5/6) + (1/6 × 1/4)
On further calculation
= 3/24 + 5/24 + 1/24
= 3/8
Hence, the probability that at least one of them is selected in 3/8.