Question

One number is selected from the first \[50\] natural numbers. What is the probability that it is the root of the inequality \[x + \dfrac{{256}}{x} > 40\]

Answer 1

Hint: The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory, which is also used in probability distribution, where you will learn the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first, we should know the total number of possible outcomes.

Complete step-by-step answer:
Sample Space: The sample pace associated with a random experiment is the set of all possible outcomes. An event is a subset of the sample space.
Event: An event E is said to occur on a particular trial of the experiment if the outcome observed is an element of the sample space.
We know that Probability (event) \[ = \dfrac{{Number\; of \;favourable\;outcomes}}{{Total\;number \;of \;outcomes}}\]
We have the inequality \[x + \dfrac{{256}}{x} > 40\]
Taking LCM we get
\[{x^2} + 256 > 40x\]
Taking all the terms on one side we get ,
\[{x^2} - 40x + 256 > 0\]
And hence we have \[(x - 32)(x - 8) > 0\]
Therefore \[x \in ( - \infty ,8) \cup (32,\infty )\]
Now , we need to select numbers from \[1\] to \[50\]. We need to have a solution from \[1\] to \[8\] and then \[32\] to \[50\].
Therefore favourable outcomes \[ = 8 + 19 = 27\]
Therefore the required probability \[ = \dfrac{{27}}{{50}}\]
The \[x \in ( - \infty ,8)\] does not satisfy the inequality \[x + \dfrac{{256}}{x} > 40\].
So the only solution is \[x \in (32,\infty )\] from \[32\] to \[50\] which are \[19\] numbers.
Therefore the required probability \[ = \dfrac{{19}}{{50}}\]
So, the correct answer is “\[ \dfrac{{19}}{{50}}\]”.

Note: The meaning of probability is basically the extent to which something is likely to happen. Remember about random experiments, sample space and favourable outcomes related to the given event. Probability of any event can be between 0 and 1 only. Probability of any event can never be greater than 1. Probability of any event can never be negative.