Probability of carrying an autosomal recessive trait given both parents are carriers
I recently came across a question that asked for the probability of having a child with an autosomal recessive trait, namely cystic fibrosis. From various parts of the question, we can conclude that the father is definitely a heterozygous carrier and that the mother's parents are both heterozygous carriers. In this situation, the only way to produce a child with cystic fibrosis is if both parents are heterozygous carriers, and this will happen with a probability of $\frac{1}{4}$.
The part that causes confusion is figuring out the probability that the mother is a carrier.
One student reasoned that a cross between two heterozygous parents will produce a 1:2:1 genotypic ratio, so $\frac{1}{2}$ of the progeny will be heterozygous carriers. Therefore, the probability that the mother is a carrier is $\frac{1}{2}$ and the probability of producing a child with cystic fibrosis is $\frac{1}{2}*\frac{1}{4}=\frac{1}{8}$
Another student also referenced the 1:2:1 ratio but reasoned that since the mother is phenotypically normal, that the ratio is actually just 1:2 because there is no chance of being homozygous recessive. 882
Therefore, the probability that the mother is a carrier is $\frac{2}{3}$ and the probability of producing a child with cystic fibrosis is $\frac{2}{3}*\frac{1}{4}=\frac{1}{6}$
These two answers are also found at this other question.
The last solution proposed is the most interesting. This student referenced the Monty Hall Problem, which is indeed somewhat related to this problem (read more about the Monty Hall Problem). Using the same ratio as above, we can conclude that there is a $\frac{1}{4}$ chance that the mother carries no alleles for cystic fibrosis and a $\frac{3}{4}$ chance that she does. The fact that she definitely does not carry two alleles does not change this probability. Therefore, the probability that the mother is a carrier is $\frac{3}{4}$ and the probability of producing a child with cystic fibrosis is $\frac{3}{4}*\frac{1}{4}=\frac{3}{16}$.
Out of these proposed solutions, which one is correct? And why are the others incorrect?
EDIT: To be clear, the solution I initially considered to be correct was the second one. 882