First slide

Introduction to probability

Question

A natural number is chosen at random from the first 100 natural numbers. The probability that $x + \frac{100}{x} > 50$ is

Moderate

A

$\frac{1}{10}$
B

$\frac{11}{50}$
C

$\frac{11}{20}$
D

none of these

Solution

We have,
$x + \frac{100}{x} > 50$
$\begin{array}{l} or x^{2} + 100 > 50 x \\ or (x - 25)^{2} > 525 \\ or x - 25 < - \sqrt{525} or x - 25 > \sqrt{525} \\ or x < 25 - \sqrt{525} or x > 25 + \sqrt{525} \end{array}$
As x is a positive integer and $\sqrt{525} = 22 .91$ , we must have x $\leq$ 2 or x $\geq$ 48. Thus, the favorable number of cases is 2 + 53 = 55.
Hence, the required probability is $\frac{55}{100} = \frac{11}{20}$

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