A breeder crosses a purebred black Lab with the genotype BBEE and a purebred yellow Lab with the genotype bbee, producing offspring that are black. A cross is done between these offspring and a Lab with the genotype bbee. What is the expected ratio of black : chocolate : yellow fur colors?

Labrador coat colors are different because of variations in two genes, namely, TYRP1 and MC1R.
TYRP1 codes for a dark pigment known as eumelanin.
Whereas, MC1R is an extension (E) gene that codes for melanocortin 1 receptor which signals pigment producing cells to produce eumelanin.
A fully functional TYRP1, represented by "B" is considered to the dominant trait, whereas the "b" is recessive.
In the MC1R, the presence of a single copy of dominant "E" will result in coat pigmentation whereas, if both recessive "e" copies are present, the lab will be yellow.

Punnet Square:

Parents: (True Black) BBEE X (True yellow) bbee

1st Filial: BbEe (All black)

2nd cross: BbEe X bbee

Gametes: BE,Be, bE, be X be,be,be,be

2nd Filial:

BE Be bE be

be BbEe Bbee bbEe bbee

Genotypes and Phenotypes in Labrador Coat Color:

Black Labs will have at least one dominant allele at both the B and E loci. This includes the genotypes: BBEE, BBEe, BbEE, or BbEe.
Chocolate Labs will have at least one dominant E allele, but must have only recessive b alleles. This includes all genotypes corresponding to bbEE and bbEe.
Yellow Labs with black skin pigment will have a dominant B allele but must have recessive e alleles: BBee or Bbee.
Yellow Labradors with no skin pigment i.e. the genotype bbee, have recessive alleles only in both loci. These are referred to as Dudleys.

Phenotypic Ratio:

Out of 16, 4 are Black, 4 are Chocolate and 8 are Yellow. Therefore, the ratio will be 4:4:8 or 1:1:2 to be exact.