Question

Two numbers are selected at random from integers $1$ through $9$. If the sum is even, find the probability that both the numbers are odd.

Answer 1

Let $A=$Getting two odd numbers, $B=$Getting the sum as an even number.

Required probability $=P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}=\frac{\frac{{}^5{C}_2}{{}^9{C}_2}}{\frac{{(}^4{C}_2{+}^4{C}_2)}{{}^9{C}_2}}=\frac{{}^5{C}_2}{{}^4{C}_2{+}^5{C}_2}=\frac{10}{6}$