Question

A number is chosen at random from among the first $50$ natural numbers. The probability that the number chosen is either a prime number or a mutiple of $5$ is

• A. $\frac{13}{25}$
• B. $\frac{12}{25}$
• C. $\frac{17}{50}$
• D. $\frac{9}{25}$

Answer 1

The correct option is B $\frac{12}{25}$

Total number of ways of selecting a number from first $50$ natural numbers is $50$
$\therefore n\left(S\right)=50$
Let $A$ be the event for prime number upto first $50$ natural numbers.
$A=\{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47\}$
$P\left(A\right)=\frac{15}{50}$
Let $B$ be the event of multiple of $5$ upto first $50$ natural numbers.
$B=\{5,10,15,20,\cdots ,50\}$
$P\left(B\right)=\frac{10}{50}$
$P(A\cap B)=\frac{1}{50}$

The favourable ways of choosing a number which is either a prime number or a mutiple of $5=P(A\cup B)$
$\Rightarrow P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)=\frac{12}{25}$