Question

A bag contains tickets numbered from 1 to 20. Two tickets are drawn. Find the probability that (i) both the tickets have prime numbers on them (ii) on one there is a prime number and on the other there is a multiple of 4.

Answer 1

Clearly, the sample space is given by S = {1, 2, 3, 4, 5,...19, 20}.
∴ n(S) = ²⁰C₂ = 190
(i)
Let E₁ be the event where both the tickets have prime numbers on them.
Then E₁ = {2, 3, 5, 7, 11, 13, 17, 19}
∴ Favourable number of ways = n(E₁) = ⁸C₂
Hence, required probability = P(E₁) = $\frac{n\left(E_1\right)}{n\left(S\right)}=\frac{{}^{8}C_2}{{}^{20}C_2}=\frac{14}{95}$

(ii)
Let E₂ be the event where one ticket has a prime number, while the other has a multiple of 4.
Then prime numbers = {2, 3, 5, 7, 11, 13, 17, 19}
and multiples of 4 = {4, 8, 12, 16, 20}
∴ Favourable number of ways, n(E₂) = ⁸C₁× ⁵C₁ = 8 × 5 = 40
Hence, required probability, P(E₂ ) = _{$\frac{n\left(E_2\right)}{n\left(S\right)}=\frac{40}{190}=\frac{4}{19}$}