Question

A two digit number is to be formed from the digits 0, 1, 2, 3, 4. Repetition of the digits is allowed. Find the probability that the number so formed is a -
(1) prime number
(2) multiple of 4
(3) multiple of 11.

Answer 1

Two digits numbers which can be formed using the digits 0, 1, 2, 3, 4 are 10, 11, 12 ,13 ,14 ,20, 21 ,22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43 and 44
Therefore, total numbers outcomes = 20
(1) Number of favourable outcomes (prime numbers) = 11, 13, 23, 31, 41 and 43
= 6
Therefore, required probability = $\frac{\mathrm{Number}\mathrm{of}\mathrm{favorable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$
$$\begin{gathered} =\frac{6}{20} \\ =\frac{3}{10} \end{gathered}$$​
Hence, the probability that the prime number formed is $\frac{3}{10}$.

(2) Number of favourable outcomes (multiple of 4) = 12, 20, 24, 32, 40 and 44
= 6
Therefore, required probability = $\frac{\mathrm{Number}\mathrm{of}\mathrm{favorable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$
​$$\begin{gathered} =\frac{6}{20} \\ =\frac{3}{10} \end{gathered}$$​
Hence, the probability that the number so formed is a multiple of 4 is $\frac{3}{10}$.
(3) Number of favourable outcomes (multiple) = 11, 22, 33 and 44
= 4
Therefore, required probability = $\frac{\mathrm{Number}\mathrm{of}\mathrm{favorable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$
​$$\begin{gathered} =\frac{4}{20} \\ =\frac{1}{5} \end{gathered}$$​
​Hence, the probability that the number so formed is a multiple of 11 is $\frac{1}{5}$.