Unfortunately, calculating probability can become confusing as more traits are considered. Joint probability is the probability of two or more events occurring together. However, whether these events are independent or not independent determines how their probabilities are calculated.
If events are independent then the probability that one event occurs is not influenced by the occurrence of a second event. For example, consider traits on two loci that are responsible for coat color. On the "B" locus a dog can be black (BB or Bb) or chocolate (bb) and on the "S" locus a dog can be solid (SS or Ss) or spotted (ss). Let's say that we want to know the probability of getting a black, spotted animal from a BbSs x BbSs cross.
First, we need to calculate the individual probabilities of each event - P(black) and P(spotted).
Probabilities of offspring phenotypes and genotypes can be calculated if the parental genotypes are known. For example, black coat color (B) is dominant to brown (b) in dogs. If we mate two heterozygous, black dogs, the probability of getting a black pup would be 1 (number of BB events) + 2 (number of Bb events) / 4 (the total number of events), or 3/4.
Using a similar analysis, the probability of getting a spotted (ss) pup from an Ss x Ss mating is 1/4 or 0.25.
Finally, the joint probability of getting a pup that is black and spotted is calculated by multiplying the individual probabilities of each event:
P(A and B both occur) = P(A) x P(B)
Thus, P(B_ss) = P(B_) x P(ss) = 3/4 x 1/4 = 3/16