# Try to answer this genetics/statistics puzzle I came up with

Hopefully this’ll be interesting to a few people. I don’t claim to have come up with a definitive answer, so the solution I spoiler in the next post is only my best guess.

Premise: In the native population of one of those islands that only appear in puzzle books, some natives have big noses and some natives have tiny noses. There’s nothing in between, so it’s pretty easy to tell which is which. Some basic testing has proved that the nose size among these natives is controlled by a single dominant/recessive gene. Everybody carries two copies of the gene, either big nose or small nose. If they have both genes, whichever is dominant controls the size of nose the native shows.

36% of the population has big noses. 64% of the population has small noses.

If a ‘mixed’ couple (one has a big nose, one has a small nose) have a baby, the odds that the baby will have a big nose is around 38%, plus or minus two percent.

Based on this, can you guess if the big nose gene or small nose gene is dominant? How certain of it are you? (Or if I messed up, does the whole story fail to hold water?)

ETA: Oh, I guess my answer assumes that there’s no bias either for people with the same nose size to marry, or to seek out somebody with a different nose size; they’re pretty agnostic about it on this island.

Here’s the answer I came up with…

[spoiler]
First, we need to know the frequency of genes in the population either way. If big noses are dominant, then only people who have two small-nose genes will have small noses. Since we know the frequency of the small-nose phenotype is 0.64, we can estimate the frequency of the small-nose genotype to be the square root of 0.64, or 0.8, and thus the frequency of the big-nose genotype is 0.2, or 20% of the genes.

Similarly, if the small-nose genes are dominant, big-nose gene frequency is the square root of 0.36, or 0.6, and small-nose genes account for the other 40%.

If one partner in a mixed-nose couple carries two copies of the dominant gene, then they’re going to pass that trait on to the baby no matter what. (Ignoring mutations.) So we need to know the likelihood that somebody showing the dominant trait is double-dominant. If the big noses are dominant, then double-dominants are 4% of the population, or 1/9th of the 36% that have big noses, or around 11% of the big noses. If the small noses are dominant, then double-dominants would be 16% of the total population, or 25% of the 64% that show small noses, one quarter.

So now we can estimate the overall probabilities for our mixed-nose couples under both hypotheses. If big noses are dominant, then the chances that the big-nose partner carries both genes is 8/9. If he does carry both genes, then s/he presumably has a 50% chance of passing the small-nose gene to the baby. The small-nose partner will certainly pass a recessive small-nose gene to the baby. Thus, the chances that the baby will have a small nose are only 4/9, or about 44%, with a nearly 56% chance that the baby will get a big schnozz. That doesn’t fit the given stat about babies of mixed-nose couples.

On the other hand, if small noses are dominant, the chances that the small-nose partner will pass on a recessive big-nose gene can be calculated the same way: 75% (the chance that the small-nose partner actually has a recessive nose gene) * 50%, or 37.5%, well within the margin of error for the 38% stat. Thus, small noses are a dominant trait.

I know I used the usual simplifications about genetics puzzles, but did I make any actual mistakes?[/spoiler]

% of population distribution is irrelevant.

If Big Noses are dominant, then a homogeneous Big Nose parent will have a Big Nose child 100% of the time, while a heterogeneous Big Nose parent will have a Big Nose child ~50% of the time with a small nose parent. There’s no way for the % to drop below 50% (other than random luck or other genetic factors) even if the entire Big Nose population is heterogeneous.

If small noses are dominant, then a homogeneous small nose parent will have a Big Nose child 0% of the time, while a heterogeneous small nose parent will have a Big Nose child ~50% of the time with a Big Nose parent. Therefore if 76% of the small nose parents are heterogeneous, then a Big Nose child would be expected 38% (0.76*0.5) of the time.