Given numbers 6,7,8,......70 form an AP with a = 6 and d = 1

Let T_n  = 70. Then,

6 + (n - 1) = 70

⇒ 6 + n - 1 = 70

⇒ n = 65

Thus, total numbers of outcomes = 65.

(i) Let E₁ be the event of getting a one digit-number.

Out of these numbers, one-digit number are 6,7,8 and 9

The number of favorable outcomes = 4

Therefore P(getting a one-digit number) = P(E₁) = \(\frac{number\, of \, outcomes\,favorable\,to\,E_1}{number \,of\, all\,possible\,outcomes}\) = \(\frac{4}{65}\)

Thus, the probability that the card bears a one-digit number is  \(\frac{4}{65}\).

(ii) Let E₁ be the event of getting a number divisible by 5.

Out of these numbers, number divisible by 5 are 10,15,20,......,70.

Given number 10,15,20,.......,70 form an AP with a = 10 and d = 5

Let T_n = 70. Then,

10 + (n - 1)5 = 70

⇒ 10 + 5n - 5 = 70

⇒ 5n = 65

⇒ n = 13

Thus, number of favorable outcomes = 13.

Therefore P(getting a number divisible by 5) = P(E₂) = \(\frac{number\, of \, outcomes\,favorable\,to\,E_2}{number \,of\, all\,possible\,outcomes}\) = \(\frac{13}{65}\) = \(\frac{1}{5}\)

Thus, the probability that the card bears a number divisible by 5 is \(\frac{1}{5}\).

(iii) Let E₃ be the event of getting an odd number less then 30.

Out of these numbers, odd number less then 30 are 7,9,11,......,29.

Given number 7,9,11,.......,29 form an AP with a = 7 and d = 2.

Let T_n = 29. Then,

7 + (n - 1)2 = 29

⇒ 7 + 2n - 2 = 29

⇒ 2n = 24

⇒ n = 12

Thus, number of favorable outcomes = 12.

Therefore P(getting a odd number less than 30) = P(E₃) = \(\frac{number\, of \, outcomes\,favorable\,to\,E_3}{number \,of\, all\,possible\,outcomes}\) = \(\frac{12}{65}\)

Thus, the probability that the card bears a odd number less than 30 is  \(\frac{12}{65}\).

(iv) Let E₄ be the event of getting a composite number between 50 and 70.

Out of these numbers, composite number between 50 and 70 are 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68 and 69.

Thus, number of favorable outcomes = 15.

Therefore P(getting a composite number between 50 and 70) = P(E₃) = \(\frac{number\, of \, outcomes\,favorable\,to\,E_3}{number \,of\, all\,possible\,outcomes}\) = \(\frac{15}{65}\) = \(\frac{3}{13}\) 

Thus, the probability that the card bears a composite number between 50 and 70 is \(\frac{3}{13}\).