An experiment has 10 equally likely outcomes. Let \(A\) and \(B\) be two 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

\(P(A)=\frac{2}{5}\)
For independent events,
\[
\begin{aligned}
& P(A \cap B)=P(A) P(B) \Rightarrow P(A \cap B) \leq \frac{2}{5} \\
& \Rightarrow P(A \cap B)=\frac{1}{10}, \frac{2}{10}, \frac{3}{10}, \frac{4}{10} \\
& \text { (i) } P(A \cap B)=\frac{1}{10} \\
& \Rightarrow P(A) \cdot P(B)=\frac{1}{10} \\
& \Rightarrow \quad P(B)=\frac{1}{10} \times \frac{5}{2}=\frac{1}{4}, \text { not possible. } \\
& \text { (ii) } P(A \cap B)=\frac{2}{10} \Rightarrow \frac{2}{5} \times P(B)=\frac{2}{10} \\
& \Rightarrow \quad P(B)=\frac{5}{10}, \text { outcomes of } B=5
\end{aligned}
\]

\[
\text { (iii) } \begin{aligned}
& P(A \cap B)=\frac{3}{10} \Rightarrow P(A) \cdot P(B)=\frac{3}{10} \\
& \Rightarrow \quad \frac{2}{5} \times P(B)=\frac{3}{10} \\
& P(B)=\frac{3}{4}, \text { not possible }
\end{aligned}
\]

\[
\text { (iv) } \begin{aligned}
& P(A \cap B)=\frac{4}{10} \\
\Rightarrow & P(A) \cdot P(B)=\frac{4}{10} \\
\Rightarrow & P(B)=1, \text { outcomes of } B=10 .
\end{aligned}
\]