An experiment has 10 equally likely outcomes. Let $A$ and $B$ be non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is

$\text{Given: } n(S) = 10 \text{ equally likely outcomes}$

$n(A) = 4, \qquad A,B \neq \varnothing$

$\text{Find } n(B) \text{ such that } A \text{ and } B \text{ are independent}$

$\text{i.e. } P(A \cap B) = P(A)\,P(B)$

$P(A) = \dfrac{4}{10} = \dfrac{2}{5}$

$\text{let } n(B) = k,\quad P(B) = \dfrac{k}{10}$

$A,B \text{ independent} \iff P(A \cap B) = P(A)\,P(B)$

$\iff P(B \mid A) = P(B)$

$P(A)\,P(B) = \dfrac{2}{5}\cdot\dfrac{k}{10} = \dfrac{k}{25}$

$P(A \cap B) = \dfrac{|A \cap B|}{10}$

$\Rightarrow |A \cap B| = \dfrac{10k}{25} = \dfrac{2k}{5}$

$|A \cap B| = \dfrac{2k}{5} \in \mathbb{Z} \ \Rightarrow\ k = 5 \ \text{or}\ 10$

$k=5:\ |A \cap B| = 2 \ (\le 4)\quad \checkmark$

$k=10:\ |A \cap B| = 4 \ (\le 4)\quad \checkmark$

$k=2,4,8 \ \text{or}\ 3,6,9 \ \text{or}\ 4,8:\ \dfrac{2k}{5} \notin \mathbb{Z}$

$n(B) = 5 \ \text{or}\ 10 \ \Rightarrow\ \text{Option } (4)$