In a haplodiploid ant species, sons are haploid and produced from unfertilized eggs, while daughters are diploid and produced from fertilized eggs. A single queen establishes a colony after mating with three different unrelated males. The queen uses the sperm from all three males equally to produce her female offspring. The average genetic relatedness between two randomly selected female offspring in this colony will be:

In a haplodiploid system, females are diploid, receiving half of their genes from the mother (queen) and half from the father (male). Males are haploid, meaning all sperm produced by a single male are genetically identical.

The genetic relatedness between two full sisters (sharing the same father) is $0.75$. This is because they share $100\%$ of their paternal genes ($0.5 \times 1 = 0.5$) and $50\%$ of their maternal genes ($0.5 \times 0.5 = 0.25$), giving $0.5 + 0.25 = 0.75$.

The genetic relatedness between two half-sisters (having different fathers) is $0.25$. They share $0\%$ of their paternal genes (since the males are unrelated) and $50\%$ of their maternal genes, giving $0 + 0.25 = 0.25$.

Since the queen mates with $3$ males and uses their sperm equally, the probability that two randomly selected daughters share the same father is $\dfrac{1}{3}$, and the probability that they have different fathers is $\dfrac{2}{3}$.

The average genetic relatedness ($r_{avg}$) is the weighted average of these probabilities:

$r_{avg} = \left(\dfrac{1}{3} \times 0.75\right) + \left(\dfrac{2}{3} \times 0.25\right)$

$r_{avg} = 0.25 + \dfrac{0.50}{3}$

$r_{avg} = 0.25 + 0.1667 = 0.4167$

Rounding to three decimal places, the average genetic relatedness is $0.417$.

Answer: $0.417$