Question

A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le 50$, then the probability of choosing $n$ is $p$, and if $n>50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is

• A. $0.05$
• B. $0.065$
• C. $0.08$
• D. $0.1$

Answer 1

Answer: (C)

$0.08$

The probability of choosing a number '$n$' such that $n\le 50$ is given as $p$. There are exactly $7$ perfect squares less than $50$. So, probability of getting a perfect square less than or equal to $50$ is $7/50$.
Similarly, the probability of choosing a number '$n$' such that $n>50$ is given as $3p$. There are exactly $3$ perfect squares from $50$ to $100$. So, the probability of getting a perfect square greater than $50$ and less than or equal to $100$ is $3/50$.
Total probability of choosing the number from '$1$' to '$100$' is $p+3p=4p$.
Probability that the number chosen is less than or equal to $50$ and a perfect square is $p\times \frac{7}{50}$
Probability that the number chosen is greater than $50$ not exceeding $100$ and a perfect square is $3p\times \frac{3}{50}$
Hence the probability that any number chosen is a perfect square is
$\frac{p\times \frac{7}{50}+3p\times \frac{3}{50}}{p+3p}$
Simplifying we get, $0.08$.