(October 21st, 2020, 16:00)thrawn Wrote: A little problem to test Omar's knowledge of them odds.
I have green eyes, my wife has brown eyes. Both her parents have brown eyes and so do her grandfathers but her grandmothers both have green eyes. We assume one gene is responsible for eye colour, brown is dominant and green recessive, and there are no mutations.
You have to specifically state that assumption, because eye color in humans doesn't actually work that way (which it sounds like you know).
The closest thing to an actual simple recessive Mendelian trait in humans I can think of, that isn't a disease/defect and is something you can expect to see commonly, is earwax consistency (dry vs. moist; the former is the recessive).
Or maybe not commonly, because how many people look at other people's earwax?
Other things that are actually real and get taught in class, like red-green color blindness and ABO blood type, already have complications (the former is X-linked; the latter has 3 alleles not 2).
(October 21st, 2020, 17:19)thrawn Wrote: Makes me think - is the werewolf gene dominant or recessive? What's it's frequency? Shall we lynch based on earwax??
Well, scum this game is supposed to be munchkins. Rather like the tabletop game Munchkin, which is close enough for our purposes.
But if you want an answer to your actual question ... for D&D 2nd edition werewolves, a male werewolf crossed with a female human always yields a human baby; at the onset of puberty, such a child has a 10% chance annually of transforming into a full werewolf, so almost all such children will eventually become true werewolves. (No it doesn't say what happens the other way around. Yes this level of detail is normal for 2nd edition.)
That suggests sort-of-Mendelian-dominance. "Sort-of-dominant" is in fact typical for lots of things that are called Mendelian dominant.
If we're assuming full-blown Mendelian dominant, and assuming there is no natural selection (this is a common simplifying assumption in biology, like in physics assuming friction doesn't exist and spherical cows) then the frequency of the villager allele is the square root of the fraction of players who are villagers. So if 75% of players are town, the villager allele frequency is sqrt(0.75) = ~0.866.
(October 21st, 2020, 16:31)AdrienIer Wrote: As a math teacher I couldn't pass an opportunity to think about probabilities. The answer is
2/3
You know what, next time one of my students finishes his or her work early, instead of finding another exercise on the theme of that day's class I'll ask them your question.
Think it through again, it's a tricky question
Can we assume that the grandfathers have a 2/3 probability of Brown-Green genes and a 1/3 probability of Brown-Brown genes? Because otherwise I'm not sure how to solve this.
Alright, I'm getting 2/3 as well. I'm not sure what I'm doing wrong. I am also assuming that the genes are passed through the Punnett square method from middle school.
Your wife's parents must have Brown-Green genes, since we know that they have brown eyes and their mothers had green eyes. The unconditional probabilities of your wife's genes are:
1/4 Brown-Brown
1/2 Brown-Green
1/4 Green-Green
Since we know that she has brown eyes, the conditional probabilities are
1/3 Brown-Brown
2/3 Brown-Green
Conditional on your wife having Brown-Brown, your child will have brown eyes. Conditional on your wife having Brown-Green, your child has 1/2 chance of brown eyes and 1/2 chance of green eyes. By Bayes rule and summing up, your child has a 2/3 chance of brown eyes.
The information that first child has brown eyes is irrelevant, since it tells us that your wife has at least one Brown gene, which we knew because she has brown eyes.
I was afraid that you were trying to trick us with the 'one of my children is a boy' paradox, but no, you identified that your first child had brown eyes.
(October 21st, 2020, 16:00)thrawn Wrote: A little problem to test Omar's knowledge of them odds.
I have green eyes, my wife has brown eyes. Both her parents have brown eyes and so do her grandfathers but her grandmothers both have green eyes. We assume one gene is responsible for eye colour, brown is dominant and green recessive, and there are no mutations.
We have one child with brown eyes. If we have another, what are the chances that it will also have brown eyes?
You must have missed the part where Omar said he majored in English Lit and minored in Art History.
The last math class I took was AP Calc in high school, and my last science class was Bio 101 my freshman year of college over two decades ago.
So I’m open to discussing 2666 or Rothko’s Seagrams Murals if you have an opinion. But this is outside my pay grade.
That said, initially I came up with 2/3 like everyone else, but I’m now thinking it’s 75%, since the information about the first child increases the chances the wife has xx, since there is a 50% possibility the x for the first child’s brown eyes came from the xx, and a 25% chance it came from either xy combo, and the 50% of the time that she has xx, the child will always have brown eyes, with an additional 12.5% for each of the 25% chances she has xy.
Completed: SG2-Wonders or Else!; SG3-Monarch Can't Hold Me; WW3-Surviving Wolf; PBEM3-Replacement for Timmy of Khmer; PBEM11-Screwed Up Huayna Capac of Zulu; PBEM19-GES, Roland & Friends (Mansa of Egypt); SG4-Immortality Scares Me
Oh no, Omar has a point. The fact that the first child has brown eyes increases the chance that your wife has Brown-Brown. I think. Back to the chalkboard...
EDIT: Do I dare build a Bayesian estimator? I'm thinking in circles here. Ack.
No, unless I've missed something, I'm overthinking things. The distribution of your second child's eye colors is determined completely by parameter p, the probability that your wife has BB (brown-brown) genes. p is not a random variable - it's fixed, and if I've done the calculations right, it's 1/3. Therefore we don't have to mess with parameter estimation.
