Question

A number is chosen at random from the set $\{1,2,3,\cdots ,2000\}$. Let $p$ be the probability that the chosen number is a multiple of $3$ or a multiple of $7$. Then the value of $500p$ is

Answer 1

$E=$ A number which is multiple of $3$ or multiple of $7$
Let $A=$ Set of numbers divisible by $3$
$\Rightarrow A=\{3,6,9,\cdots ,1998\}$
$\therefore n\left(A\right)=666$
$B=$ Set of numbers divisible by $7$
$\Rightarrow B=\{7,14,21,\cdots ,1995\}$
$\therefore n\left(B\right)=285$
$A\cap B=\{21,42,63,\cdots ,1995\}$
$\therefore n(A\cap B)=95$
Now,
$n\left(E\right)=n\left(A\right)+n\left(B\right)-n(A\cap B)$
$\Rightarrow n\left(E\right)=666+285-95$
$\Rightarrow n\left(E\right)=856$
$\therefore$ Required probability
$p=\frac{856}{2000}$
$\therefore 500p=\frac{856}{4}=214$