First, let's determine the possible gametes for each parent based on their genotypes:
The female snake is homozygous recessive for the r and s traits, so her gametes will all carry the recessive alleles: rs.
The male snake is homozygous dominant for the R and S traits, so his gametes will all carry the dominant alleles: RS.
Now, let's determine the possible genotypes of the offspring when these gametes combine:
Since the two traits assort independently, we can use the product rule to calculate the probability. The probability of obtaining an offspring that is respectful (r) and sincere (s) would be the product of the probabilities of each trait.
For the respectful trait:
Probability of inheriting r from the female = 1 (since she is homozygous recessive for r)
Probability of inheriting R from the male = 1/2 (as he could pass on either R or S)
So, the probability of inheriting r = 1 * 1/2 = 1/2.
For the sincere trait:
Probability of inheriting s from the female = 1 (since she is homozygous recessive for s)
Probability of inheriting S from the male = 1/2 (as he could pass on either R or S)
So, the probability of inheriting s = 1 * 1/2 = 1/2.
Now, we can calculate the probability of having an offspring that is respectful and sincere:
Probability of being respectful and sincere = Probability of being r * Probability of being s = 1/2 * 1/2 = 1/4.
Therefore, the probability that they will have an offspring that is respectful and sincere is 1/4.