Question

Two digit number are formed from the digits 0,1,2,3,4 where digits are not repeated. Find the probability of the events that
(i) the number formed is an even number.
(ii) the number formed is greater than 40.
(iii) The number formed is prime number.

Answer 1

Two-digit numbers are formed using the digits 0, 1, 2, 3 and 4 (without repeating the digits).
We have:
Sample space = {10, 12, 13, 14, 20, 21, 23, 24, 30, 31, 32, 34, 40, 41, 42, 43}
∴ n(S) = 16

(i) Let X be the event that the number formed is an even number.
X = {10, 12, 14, 20, 24, 30, 32, 34, 40, 42}
∴ n(X) = 10
Thus, we have:
$$\begin{gathered} \mathrm{P}\left(\mathrm{X}\right)=\frac{n\left(\mathrm{X}\right)}{n\left(\mathrm{S}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{X}\right)=\frac{10}{16}=\frac{5}{8} \end{gathered}$$

(ii) Let Y be the event that the number formed is greater than 40.
Y = {41, 42, 43}
∴ n(Y) = 3
Thus, we have:
$$\begin{gathered} \mathrm{P}\left(\mathrm{Y}\right)=\frac{n\left(\mathrm{Y}\right)}{n\left(\mathrm{S}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{Y}\right)=\frac{3}{16} \end{gathered}$$

(iii) Let Z be the event that the number formed is a prime number.
Z = {13, 23, 31, 41, 43}
∴ n(Z) = 5
Thus, we have:
$\mathrm{P}\left(\mathrm{Z}\right)=\frac{n\left(\mathrm{Z}\right)}{n\left(\mathrm{Z}\right)}$
$\Rightarrow P(\mathrm{Z})=\frac{5}{16}$