As a rather weird side effect of email not working, I found myself browsing through my old website. And found the following:
Genetics on Ararat
In order to avoid the nuclear fallout from world war III, Mr Noah took his wife and the four little Noahs onto the spaceship Ark. As his purpose was the preservation of the human race he also chose four of his children’s playmates who would eventually become his sons and daughters in law. After many years floating around in space they landed on the planet Ararat, held a quadruple wedding and settled down to colonize the planet.
One of the strange things about life on Ararat was that every family always had two boys and two girls. Soon there were sixteen healthy little Araratians making the planet a joyful place. As they reached their teens it was agreed that they should be allowed to choose their own marriage partners from amongst their cousins provided they did so in such a way that no-one got left with a brother or sister. In successive generations the brother/sister taboo remained but no differentiation was made between degrees of cousinship.
Now all ten of the original settlers had brown eyes as did all the first generation Araratians. What was not known at the time was that whilst Mr Noah and his four in-laws each carried two genes for brown eyes (BB) Mrs Noah carried a recessive gene for blue eyes (Bb). According to Mendelian laws this recessive gene has a 50% chance of being passed to each of the carrier’s offspring. As this is Ararat, we can assume that the same influence that produces uniform families will distribute the gene to one son and one daughter of each affected family.
Eventually two carriers will marry and, to the great consternation of the whole community, produce a blue eyed baby.
What is the probability of the first blue eyed baby being:
a second generation Araratian?
belonging to each successive generation?
In which generation does it become more likely that there will be one or more blue eyed people than none?
I made up the story when I was attempting to teach myself statistics and got a bit sidetracked with an interest in genetics. But I never managed to solve it. I seem to recall covering sheets and sheets of paper with workings that got longer and longer but the business about brothers and sisters makes it a whole lot more complicated than just saying how many different ways can you match eight girls with eight boys.
Any mathematicians out there who can throw any light on the problem?