Question

Natural numbers from $ 51 $ to $ 150 $ are written on $ 100 $ cards. $ A $ card is drawn randomly from this set of $ 100 $ cards. What is the probability that the number written on the card drawn will either be a perfect square or a perfect cube?

• A. $ \frac{1}{25} $
• B. $ \frac {3}{50} $
• C. $ \frac{2}{25} $
• D. $ \frac{1}{20} $

Answer 1

Answer: (B)

Perfect square numbers from $51$ to $150$ are $64, 81, 100, 121, 144$. So, there are $5$ perfect square numbers from $51$ to $150$
Perfect cubes from $51$ to $150$ are $64, 125$. So there are $2$ perfect cubes from $51$ to $150$
Let A be the event that number drawn is perfect square and B be the event that number drawn is perfect cube.
$\therefore P\left(A\right)=\frac{5}{100}$ and $ P\left(B\right)=\frac{2}{100}$
Also, $P\left(A\cap B\right)=\frac{1}{100}$
Required probability $=P\left(A \cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$
$=\frac{5}{100}+\frac{2}{100}-\frac{1}{100}$
$=\frac{6}{100}=\frac{3}{50}$