4. In guinea pigs, rough coat is dominant $$ (\mathrm{R}) $$ to smooth coat $$ (\mathrm{r}) $$, and black coat $$ (\mathrm{B}) $$ is dominant to white coat (b). A rough black guinea pig bred with a rough white one produces 29 rough black offspring, 31 rough white offspring, 11 smooth black offspring, and 9 smooth white offspring. Give the genotypes of the parents, the probability for the offspring, and the actual number of each kind of offspring you would expect. If you prefer, you can insert text boxes with alleles

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Let the alleles be defined as follows:

- For coat texture: $$R$$ (rough) is dominant to $$r$$ (smooth).  
- For color: $$B$$ (black) is dominant to $$b$$ (white).

Because both parents have rough coats and produce some smooth offspring, each must have at least one $$R$$ and one $$r$$, so both are heterozygous:  
$$
\text{Parent coat genotype: } Rr.
$$

Since the rough white parent shows a white coat, its color genotype must be homozygous recessive:  
$$
\text{White phenotype: } bb.
$$

The rough black parent shows a black coat. Although it could be either $$BB$$ or $$Bb$$, the presence of white offspring in the cross means that the black parent must carry one recessive $$b$$ allele (so that it can pass $$b$$ to offspring). Therefore, its color genotype is $$Bb$$.

Thus, the parental genotypes are:
- Rough black: $$RrBb$$
- Rough white: $$Rrbb$$

Now, let’s find the probability for each kind of offspring.

1. **Coat Texture Cross ($$Rr \times Rr$$):**  
   The Punnett square for coat texture gives:  
   - $$RR$$: $$1/4$$ (rough)  
   - $$Rr$$: $$1/2$$ (rough)  
   - $$rr$$: $$1/4$$ (smooth)  
   
   So, the probability for a rough coat is $$3/4$$ and for a smooth coat is $$1/4$$.

2. **Color Cross ($$Bb \times bb$$):**  
   The Punnett square for color gives:  
   - $$Bb$$: $$1/2$$ (black)  
   - $$bb$$: $$1/2$$ (white)

3. **Combined Cross:**  
   The two traits segregate independently so we multiply the probabilities:
   - **Rough black:** $$ \left(\frac{3}{4}\right) \times \left(\frac{1}{2}\right) = \frac{3}{8} $$
   - **Rough white:** $$ \left(\frac{3}{4}\right) \times \left(\frac{1}{2}\right) = \frac{3}{8} $$
   - **Smooth black:** $$ \left(\frac{1}{4}\right) \times \left(\frac{1}{2}\right) = \frac{1}{8} $$
   - **Smooth white:** $$ \left(\frac{1}{4}\right) \times \left(\frac{1}{2}\right) = \frac{1}{8} $$

Given a total of 80 offspring, the expected numbers are:
- Rough black: $$80 \times \frac{3}{8} = 30$$
- Rough white: $$80 \times \frac{3}{8} = 30$$
- Smooth black: $$80 \times \frac{1}{8} = 10$$
- Smooth white: $$80 \times \frac{1}{8} = 10$$

The reported numbers (29 rough black, 31 rough white, 11 smooth black, and 9 smooth white) deviate slightly from these expectations due to random sampling error, which is common in actual biological crosses.

Thus, the final answers are:

- Parental genotypes:  
  Rough black: $$RrBb$$  
  Rough white: $$Rrbb$$

- Offspring probabilities:  
  Rough black: $$\frac{3}{8}$$  
  Rough white: $$\frac{3}{8}$$  
  Smooth black: $$\frac{1}{8}$$  
  Smooth white: $$\frac{1}{8}$$

- Expected numbers in 80 offspring:  
  Rough black: 30  
  Rough white: 30  
  Smooth black: 10  
  Smooth white: 10
Parental genotypes: - Rough black: $$RrBb$$ - Rough white: $$Rrbb$$ Offspring probabilities: - Rough black: 37.5% - Rough white: 37.5% - Smooth black: 12.5% - Smooth white: 12.5% Expected numbers in 80 offspring: - Rough black: 30 - Rough white: 30 - Smooth black: 10 - Smooth white: 10

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