Final answer:
The allele likely responsible for the death of one mouse despite identical environmental conditions is a recessive lethal allele, which is lethal only when two copies are present.
Step-by-step explanation:
If the two male mice are homozygous for the same allele and raised under identical conditions, yet one mouse dies while the other survives, the type of allele likely responsible for this death is a recessive lethal allele. A recessive lethal allele is only lethal when homozygous, meaning the individual has two copies of this allele. Individuals with a single copy of this allele (heterozygotes) act as carriers and typically do not show the lethal phenotype. The presence of a functional wild-type allele masks the nonfunctional allele's effects in the carriers. This phenomenon explains why death occurs only in individuals that are homozygous recessive for the allele.
Conversely, a dominant lethal allele is lethal both in the homozygous and heterozygous state, but is very rare in populations as such alleles tend to not be transmitted through generations, unless the lethality is delayed until after the reproductive age, as with Huntington's disease in humans. Because both mice were homozygous but only one died, we can infer that the allele responsible is likely to be a recessive lethal allele, not a dominant lethal allele.