Step 1: Understand the setup.
In haplodiploid ants, daughters are diploid and get half their genes from the mother and half from a father. The queen mated with three unrelated males and uses their sperm equally, so a random daughter has one of three possible fathers.
Step 2: Split relatedness into two halves.
Pick two random daughters. Their relatedness comes from the mother's side and the father's side. Each side contributes half the genome, so we average the two contributions.
Step 3: Relatedness through the mother.
The mother is diploid and passes a random one of her two gene copies to each daughter. Two daughters share the same maternal gene copy with probability $\tfrac{1}{2}$. The maternal half therefore contributes $\tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}$.
Step 4: Relatedness through the father.
First, the two daughters must share the same father. With three equally used fathers, that chance is $\tfrac{1}{3}$. If they do share a father, he is haploid and gives every daughter an identical copy, so relatedness through him is full ($1$). The paternal half contributes $\tfrac{1}{2} \times \tfrac{1}{3} \times 1 = \tfrac{1}{6}$.
Step 5: Add the two halves.
\[ r = \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \]
Step 6: Convert to a decimal.
\[ \frac{5}{12} \approx 0.417 \]
\[ \boxed{0.417} \]