Conundrum: Blue Gene Baby
I had the pleasure of observing a science teacher teach a fantastic lesson on genetics last week, and it got me thinking about the mathematics behind eye color. This Conundrum will be purely a probability question (two, actually), so I apologize in advance for over-simplifying the science.
Assume that everyone has two genes that determine eye color. For the sake of the math, we will stipulate that each gene must be either brown or blue. An individual inherits one gene from each parent. A parent will pass on one of his or her own two genes with equal probability.
Brown is dominant, which means that if an individual has one brown gene and one blue gene, then the individual will have brown eyes. An individual will also have brown eyes if both genes are brown. Only an individual with two blue genes will have blue eyes.
Now imagine this hypothetical scenario: Susan and David are a married couple, and both have brown eyes. David’s father had blue eyes, and his mother had brown eyes. Susan’s parents both had brown eyes, but her brother Bill has blue eyes. Susan and David are expecting their first child, baby Jason.
Question 1: What are the chances that Jason will have blue eyes?
Question 2: Suppose Jason had brown eyes. Susan and David are now expecting a second child, baby Ian. What are the chances that Ian will have blue eyes?
UPDATE: Both questions answered correctly by Micah. See comments for answers and discussion.
December 23rd, 2008 at 12:35 pm
My head hurts.
December 24th, 2008 at 12:39 am
Nice problem! Conditional probability is fun.
David carries the blue-eyed gene (he got it from his mom). Susan’s parents both carry the blue-eyed gene (they both gave it to her brother). This means that Susan has a 2/3 chance of carrying the gene, given that she has brown eyes.
So:
A) There’s a 1/3 chance that Susan doesn’t carry the blue-eyed gene.
B) There’s a 2/3 chance that she does.
If Susan carries the blue-eyed gene, the children have a 1/4 chance of being blue-eyed. So we can expand the above to:
A) 1/3 chance that Susan doesn’t carry the gene, so all her children are definitely brown-eyed.
B1) 1/2 (=2/3*3/4) chance that Susan carries the gene, but Jason has brown eyes anyway.
B2) 1/6 chance that Jason has blue eyes.
If Jason is brown-eyed, we’re in either situation A or B1, so there’s a 2/5 (=1/3 divided by 5/6) chance that Susan doesn’t carry the gene, and a 3/5 chance she does. So the odds that Ian has blue eyes are 3/5*1/4=3/20.
December 24th, 2008 at 2:27 am
micah, i was with you all the way to B2… i laid it out a little differently, with the same results.
i thought the 2nd question was a trick question, though, cuz i thought that the eyes of the 2nd kid weren’t dependent on the 1st kid’s eyes. although now that i think about it, i can see that if jason’s eyes were blue, that would change the odds of the 2nd kid’s eyes… that’s a good answer you got there =)
December 24th, 2008 at 3:02 am
Well played. Micah, you are correct on both questions (or at least you got the same answers as I did).
The first question is a little tricky, as it is tempting to think of it being equally likely that Susan is Brown-Blue than that she is Brown-Brown. But Blue-Brown is also a possibility that must be counted separately.
The second question is much more difficult, and I think there are a number of ways you could go wrong. On the surface, as Kimi points out, it seems that Jason’s brown eyes give us no new information. But while it is true that Ian’s eyes are not dependent on Jason’s, Jason’s brown eyes give us a bit more information about Susan. With each brown-eyed child she has without a blue-eyed one, it becomes slightly more likely that she is Brown-Brown.
David, for example, is one of five children, all of whom have brown eyes. So there is a 32/33 chance that his mother was Brown-Brown… assuming that a brown-eyed person in the general population is equally likely to be homozygous as heterozygous. If it’s actually 2/3, as above, then the chances of her being Brown-Brown would be 32/34, or 16/17.
As you might suspect by now, this story is entirely true, and I am Bill. According to Susan, Jason’s eyes are actually hazel-green and Ian’s eyes are brown.
And finally… Welcome, Little Fish! I know this sort of thing isn’t everyone’s cup of tea, but I’m sure you can appreciate the avuncular affection in trying to guess my nephews’ eye color before they were born.
Answers
Question 1: 1/6
Question 2: 3/20