If two digit numbers are made with the digits $2, \; 3, \; 5$, what is the probability (when repetition of digits is not allowed) that the number is
- greater than $35$;
- a multiple of $2$;
- a prime number.
Sample space $= S = \left\{23, \; 25, \; 32, \; 35, \; 52, \; 53 \right\}$
Number of elements in sample space $= n \left(S\right) = 6$
Let $A =$ event of having a two digit number greater than $35$
Then $A = \left\{52, \; 53 \right\}$
Number of elements in $A = n \left(A\right) = 2$
Probability of $A = P \left(A\right) = \dfrac{n \left(A\right)}{n \left(S\right)} = \dfrac{2}{6} = \dfrac{1}{3}$
Let $B =$ event of having a two digit number which is a multiple of $2$
Then $B = \left\{32, \; 52 \right\}$
Number of elements in $B = n \left(B\right) = 2$
Probability of $B = P \left(B\right) = \dfrac{n \left(B\right)}{n \left(S\right)} = \dfrac{2}{6} = \dfrac{1}{3}$
Let $C =$ event of having a two digit number which is a prime number
Then $C = \left\{23, \; 53 \right\}$
Number of elements in $C = n \left(C\right) = 2$
Probability of $C = P \left(C\right) = \dfrac{n \left(C\right)}{n \left(S\right)} = \dfrac{2}{6} = \dfrac{1}{3}$