Given the sum of two non negative integers as $100$, the find the probability that their product is greater that or equal to $\frac{18}{25}$times their greatest possible product.
Greatest possible product is $2500$. So if numbers are $x$ and $y$, then we need $xy \geq 1800$ but how are we supposed to calculate favorable and total cases?
On
The total cases are $$ x = 0, y = 100\\ x = 1, y = 99\\ \vdots\\ x = 100, y = 0 $$ For favourable cases, you just need to count how many of these have $xy\geq 1800$. It can be nice to know that the closer $x$ and $y$ are to one another, the larger their product becomes. This way you only have to test a fraction of the total cases to find all the favourable ones.
Let numbers be $50-x$ and $50+x$ so product of number will be $(50-x)(50+x)$. This is a decreasing function, maximum at $x=0$. So just solve $(50-x)(50+x)\ge 1800$
$$x^2 \le 2500-1800 = 700$$ This gives us $x\in [0,26]$ as $x$ is integer.