Gingers, Genes, and the Binomial Function

Happy Birthday, Ansley! For your birthday, I'm going to answer your question! Ansley asks: "I would like to know how probable it is that my soulless kind (aka redheads) are actually going to die out in the current century."

Would you like to help Ansley stave off the extinction of her people?

The short answer is: probably not. That link goes to a Snopes article about blondes, who suffer from similar speculation about when they're going to "die out." The conclusions are equally applicable to redheads. The thing about genes is that they exist to copy themselves-- it's literally all they do. That quality makes them hardy little bastards who don't give up without a fight.

Red hair is a recessive trait. Only about 2% of the world's population expresses the red hair phenotype, but a far greater percentage possess the genes to create it. Even if there came a time when no redheads existed on the face of the earth, it's probable that ginger genes would still be hanging around, hiding out in some brunette's gonads, ready to spring into action one or two generations down the line. As long as humans are still making babies, at least a couple of those babies are likely to be gingers.

For example, neither I nor my parents are redheads, but my grandfather was a redhead and my brother sports fashionable russet locks, so the recessive genes for redheaddedness are definitely sneaking their way through my family tree. Both my parents obviously carry them. It's possible that I carry them myself. How likely would it be that I could produce a redheaded child? Let's find out!

From left to right: me, my dad, and my redheaded brother

I must ask the forgiveness of my readers who are better-versed than I in matters of genetics, because I'm about to steamroll all over the subject's complexity in the interest of throwing around numbers and probability.

According to the Wikipedia entry on human hair color, pheomelanin is one of the major players in creating the orange and yellow hues that characterize red hair. There are two main genes involved in controlling hair color: one with a dominant brown allele and recessive blond allele, and one with a dominant pheomelanin-suppressor allele and non-dominant non-suppressor allele. For simplicity's sake, I'm going to label the first pair B (for brown) and b (for blonde), and the second one P (for no pheomelanin) and p (for pheomelanin).

My hair is dark brown, so my possible genotypes are PPBB, PPBb, PpBB, or PpBb. Let's assume that my goal is to produce at least one redheaded child. While a great number of individuals possess the recessive genes for redheadedness, my best bet is to pick a redheaded male with whom to reproduce-- he's guaranteed to have the genes I need.

A possible candidate?

The possible genotypes for my redhead sperm donor wouldn't include the P allele, since that would supress the pheomelanin responsible for red hair. The B or b alleles don't directly affect red coloring in the hair, but I feel like a ppbb individual would express stawberry-blonde hair instead of the traditional ginger we're looking for, so I'm going to exclude that genotype from the list of possibles. That narrows it down to two possible donor genotypes: ppBb and ppBB. Conveniently, these are also the target genotypes for my hypothetical redhead child.

Can you guess what time it is? It's Punnet Square Time!

Gettin' down with Gregor! This should probably be a picture of Reginald Punnett instead.
 

There's eight possibilities for the genetic mashup that would ensue after I make sweet, sweet love to my hypothetical redhead. Four of those cases are non-starters-- if it's PPBB/ppBb, PPBb/ppBb, PPBb/ppBB, or PPBB/ppBB, I can only offer my child the dominant pheomelanin-suppressor gene, dashing their Anastasia cosplay dreams in utero. Nothing I can do about that. But! If my genotype is PpBb or PpBB, both the father and I can provide the necessary p allele to our offspring, giving them the opportunity for pheomelanin-rich red hair.

For a PpBb/ppBB pairing, our offspring have a 50% chance of red hair:

 

The results are similar for PpBB/ppBB:

 

 

 And guess what? They're very much the same for PpBB/ppBb!

 

Things get interesting when it's PpBb/ppBb-- we get a couple bb-blondes.

So! Each of the eight possible parental phenotype mashups produces sixteen possible genotype outcomes for a total of 128 theoretical offspring. Of those offspring, only 30 have red hair. Thirty redheads out of 128 genotypes gives me about a 23% chance to make a ginger baby, provided I hook up with a redhead. If anybody's interested, I also made a giant probability tree with each possible genotype, mutliplied out all the probabilities, and added together the resulting redheads-- still 15/64, or 23%.

 

But a 23% chance isn't good enough for me. I wanna be, like, 95% sure that I have a redhead baby. So I can use the Binomial probability function!

 

Without getting into intense details, the Binomial function is useful in situations where there are two outcomes (in our case, a baby does or does not have red hair) and we're allowed to conduct as many trials (e.g., make as many babies) as we please. The Binomial function helps us figure out how probable it is that any number of trials in the total will be successful (e.g., if I have five kids, how likely is it that at least two of them will be ginger? More than two? At least three? Exactly one?) It's a handy, flexible tool, but the calculations are tedious to do by hand, and the applet in the link above makes things a lot easier. Thank you, Texas A&M University!

 

The two essential variables for any Binomial calculation are p, the probability of success in any individual trial, and n, the number of trials we intend to conduct. There's a 0.23 chance that any individual baby of mine will be a redhead, so let's enter 0.23 for p. I want at least one redhead child, so we select "at least" from the drop down menu and enter 1 in the box beside it. Now, we can adjust n, the number of children I'm theoretically having, and get different probabilities based on projected family sizes.

 

In a 3-child family, there's only a 54% probability that I'll get a redhead. Not enough! At 4 children, we get a 65% chance for one of them to be a redhead. Clearly, reasonable family sizes aren't going to cut it. WE NEED MORE BABIES!

 

 

Finally, at 12 trials, we get a 95.6% probability of at least one success given a 0.23 probability of success on any given trial. So me and my redhead babydaddy have to pop out a dozen babies to be (pretty much) certain of producing a redhead. There's a joke about Irish families or Modest Proposals in here somewhere. Of course, this figure isn't exactly accurate, since I don't really have a 23% chance each time I reproduce to pass on the right genes for a redhead. My genotype, though it is unknown to me, is set in stone. Remember, there's a 50% chance that I don't even have the p allele needed to make a redhead baby, and that all this energetic baby-making is for naught.

 

Now my search history includes "pile of babies."

But fear not, Ansley. Even if not a single one of my million babies has red hair, every single one of them carries a recessive, pheomelanin-friendly p allele from their father. They can still pass the ginger genes on to their children, and theoretically produce redhead grandbabies-- much in the way my red-haired grandfather had no red-haired children, but wound up with two red-haired grandchildren. While red hair is uncommon, it's virtually impossible to eradicate. So rejoice, Ansley! Your rosy locks are in no danger of vanishing from the earth any time this century, nor for many centuries hence. In fact, that red hair might increase your chances of passing on your genes...

 

Go read Saturday Morning Breakfast Cereal. It will make you happy.