Final answer:
The probability that the couple's first child will be affected by the X-linked recessive Duchenne muscular dystrophy is 1/8, considering the woman's potential carrier status and the equal chance of having a son.
Step-by-step explanation:
The probability that the couple's first child will be affected by Duchenne muscular dystrophy, which is an X-linked recessive disease, depends on whether the woman is a carrier of the disease. Since the woman's brother has the disease, and knowing that mothers pass one of their X chromosomes to each child, the woman has a 50% chance of being a carrier herself (since her father would have a normal Y chromosome).
Now, for their sons, there is a 50% chance the son will inherit the affected X chromosome from his mother if she is indeed a carrier. No daughters would be affected, only potentially carriers, as they would inherit a normal X chromosome from their father. So, the calculation for the probability that the first child is affected is as follows: there's a 50% chance the child will be a son and, given that, another 50% chance that the son would inherit the affected gene. This yields a 1/4 chance a son would be affected. Since there's only a 50% chance the first child will be a son, these probabilities are multiplied together: ½ * ¼ = ½ * ½ = ¼ or 1/8. Therefore, the probability of an affected first child is 1/8.