In Labrador retrievers, black coat is dominant to chocolate, normal vision is dominant to progressive retinal atrophy (PRA), and normal hip joint is dominant to hip dysplasia. The alleles for all of these traits assort independently. Two dogs that are heterozygous for alleles of all three genes are crossed. Using the rules of probability (not a Punnett square), what is the chance that the first pup born to these dogs will be black, have PRA, and have hip dysplasia

3/64 B- vv dd

Explanation:

We know that the alleles for all of the three traits assort independently. So we can think about them separately.

Coat color:

Let us say that the dominant allele coding for Black is B while the recessive allele coding for chocolate is b.

Parentals) Bb  x  Bb

F1) 1/4 BB, 2/4 Bb, 1/4 bb

Vision:

Let us say that the dominant allele coding for normal vision is N while the recessive allele coding for PRA is n.

Parentals) Nn   x   Nn

F1) 1/4 NN, 2/4 Nn, 1/4 nn

Hip joint

Let us say that the dominant allele coding for the normal hip joint is D while the recessive allele coding for dysplasia is d.

Parentals) Dd   x   Dd

F1) 1/4 DD, 2/4 Dd, 1/4 dd

Now, we need to know the chance that the first pup born to these dogs will be black, have PRA, and have hip dysplasia. The genotype of this individual would be B- nn dd, where the symbol - represents the dominant and/or the recessive allele.

Black coat = 1/4 BB + 2/4 Bb = 3/4 B-

PRA = 1/4 nn

Dysplasia = 1/4 dd

The probability of getting the first individual with these genotypes results from their multiplication.

3/4 B- x 1/4 vv x 1/4 dd = 3/64 B- vv dd

The product probability rule allows us to calculate the probability of occurrence of event A and event B at the same time. It is about a joint probability of two or more events that might happen simultaneously, not excluding each other. This rule is based on the dependence or independence of the events. Two events A and B are independent of each other if one of them does not affect the occurrence of the other one. For independent events, the rule establishes that, given two different events A and B, the probability of occurrence of both together is  

P(A∩B) = P(A) x P(B)