We are given \(P(E) = 0.1\), \(P(F) = 0.3\), and \(P(E \cup F) = 0.4\). We need to find \(P(F|E)\).

First, we use the formula for the probability of the union of two events:

\(P(E \cup F) = P(E) + P(F) - P(E \cap F)\)

Plugging in the given values:

\(0.4 = 0.1 + 0.3 - P(E \cap F)\)

Solving for \(P(E \cap F)\):

\(P(E \cap F) = 0.1 + 0.3 - 0.4 = 0\)

Now, we use the formula for conditional probability:

\(P(F|E) = \frac{P(E \cap F)}{P(E)}\)

Plugging in the values:

\(P(F|E) = \frac{0}{0.1} = 0\)