Question

Let us assume that the sum of two positive numbers is $100$. Choose the correct probability that their product is greater than $$1000$$ from the following options.
1) $\dfrac{7}{9}$
2) $\dfrac{7}{{10}}$
3) $\dfrac{2}{5}$
4) None of these.

Answer 1

Hint: To solve this problem we will use some inequalities namely the A.M-G.M rule. In this rule, we can get the relation between the sum and product. By using the relation we can determine the number of outcomes for the product. So that we can calculate the probability.

Complete step-by-step solution:
Given that,
Sum of two positive numbers $ = 100$,
Let us take the positive numbers as $x,y$.
$ \Rightarrow x,y > 0$, and
$ \Rightarrow x + y = 100$.
We know that,
A.M-G.M rule for two positive numbers $a,b$ is
$\dfrac{{a + b}}{2} \geqslant \sqrt {ab} $.
By applying the above rule to the positive numbers $x,y$. We get,
$\dfrac{{x + y}}{2} \geqslant \sqrt {xy} $
We have $x + y = 100$, by substituting the value of $$x + y$$ in the above inequation. We get,
$\sqrt {xy} \leqslant 50$
By squaring on both sides, we get
$xy \leqslant 2500$
Therefore the number of outcomes for the product is 2500 since $x,y > 0$.
Now we have to find the favorable outcomes to find the required probability.
Let us take the event, $E = xy > 1000$.
$n\left( E \right) = $number of favorable outcomes $ = [1001,2500]$.
We know that,
Probability, $P\left( E \right) = \dfrac{{n\left( E \right)}}{{T\left( E \right)}}$. Where, $T\left( E \right)$ is the total number of outcomes.
We have $$T\left( E \right) = [1,2500],n\left( E \right) = = [1001,2500]$$.
The required probability, $P\left( E \right) = \dfrac{{1500}}{{2500}}$ (we use the length of the interval to determine the probability since the whole interval is in favorable and total outcomes)
$ \Rightarrow P\left( E \right) = 0.6$.
Therefore, The required probability is $0.6$.
The correct option is 4.


Note: There is another way of doing this problem if the given question is about integers. By taking numbers as $x,100 - x$ and forming a quadratic inequation and finding the total outcomes. Many students do the same thing for this question but that is wrong since the question is asked for positive numbers, not for positive integers.