There seems to be considerable confusion as to the inheritance of horns or the lack thereof in Hereford cattle.  This is, in my opinion, due mostly to the champions of Polled Herefords who list their evidence based on the history of the Polled Hereford in America – or a version of history taken slightly out of context and made to sound somewhat feasible to the uninformed.  Their version of history goes something like this:In winter of 1901 Warren Gammon, credited as being the father of American Polled Herefords, sent out inquiries to each of the some 2,500 members of the American Hereford Cattle Breeders Association (now simply the American Hereford Association) asking if they’d ever had a purebred Hereford calf that had failed to develop horns.  Through the replies received, Gammon located 10 hornless cows and 4 bulls.  Shortly after, Gammon located another 2 bulls, and the resulting 16 head were the foundation for Polled Herefords today.
This sounds logical at first glance.  Let me direct your attention to a few more facts, which in my opinion, shakes things up a little:
1- In “The Hereford in America” by Donald Ornduff (which, by the way, is my source for all the above information – very good reading, I recommend it highly) there is a table listing registrations and transfers for the years 1907-1956.  Unfortunately, the book does not list the registrations received in the year 1901, but for the sake of my argument 1907 is close enough.  In the year 1907: 29,089 animals were recorded (registered) in the AHA herd book while 21,350 were transferred.  29,000 animals registered in 1907!  That’s a lot.
2- In “The Battle of Bull Runts” by L.P. McCann, we learn that there is an estimated 5% degree of error in Hereford registrations.  Assume for a moment that figure is remotely accurate, and assume it was the same in 1901 as it was in the 1950′s.  Logical assumptions, right?
5% of 29,000 is 1450 head of cattle registered incorrectly in the year 1907.  So, subtract a few hundred because there were probably less Herefords in 1901.  You still have a heck of a lot more incorrectly registered animals that the measly 16 Warren Gammon started with!!
My conclusions are obvious, I’ll let you draw your own from the information I’ve put together.  Read the two books I quote.  I didn’t make that information up, and the authors of those two books were well respected within AHA for years – they knew their stuff.
All this is not to put down Polled Herefords in any way.  They are excellent animals, and a very logical choice for highly populated areas or as pets for children.  After over 100 years of breeding them as consistently as horned Herefords, there is no longer any genetic difference worth mentioning other than the lack of horns.
My point is simply this:
A HORNED HEREFORD COW MATED TO A HORNED HEREFORD BULL
WILL NEVER PRODUCE A POLLED CALF.
If you get a polled calf out of a horn to horn mating, someone’s bull jumped your fence when you weren’t looking.  For further evidence, please read the scientific article below:

Inheritance of Polledness, Horns and Scurs in Beef Cattle
——————————————————————————–B. C. Allison
——————————————————————————–In beef cattle of European ancestry the trait of being polled or having horns is determined by one pair of genes. One gene in the pair is inherited from the dam and the other from the sire. The polled gene (P) is dominant to the horned gene (p). If an animal has two polled genes (PP), homozygous, or one polled and one horned gene (Pp), heterozygous, it will be polled. However, if it is heterozygous polled (Pp) it may pass either the polled or horned gene on to its offspring. The only situation when an animal will be horned is when it possesses two recessive horned genes (pp), homozygous horned. Table 1 illustrates the expression of polledness or horns and what genes and traits can be expected to be passed to the offspring from the various matings.
If an animal of European breeding, not of Zebu ancestry, has horns then you can determine from visual observation that it is homozygous for the recessive horned gene. However, if the animal is smooth polled or scurred it is impossible to determine from visual observation if it is genetically homozygous polled (PP) or heterozygous polled (Pp). The homozygous polled bull with two polled genes will sire only polled calves. Only through the offspring produced can the number of polled genes be determined. The best test of a bull is to mate the polled bull in question to horned cows. A polled bull bred to horned cows that produce one or more horned calves is heterozygous polled (one recessive gene for horns), regardless of how many polled calves are produced. Table 2 gives the probability of a polled bull being homozygous polled if no horned calves are produced from matings with horned cows.
Table 1. Genetic Expression of Polledness or Horns and Expected Inheritance by Offspring Sire Dam Calves Homozygous polled (PP) Homozygous polled (PP) 100% Homozygous polled (PP) Homozygous polled (PP) Heterozygous polled (Pp) 50% Homozygous polled (PP)     50% Heterozygous polled (Pp) Homozygous polled (PP) Homozygous horned (pp) 100% Heterozygous polled (Pp) Heterozygous polled (Pp) Homozygous horned (pp) 50% Heterozygous polled (Pp)     50% Homozygous horned (pp) Heterozygous polled (Pp) Heterozygous polled (Pp) 25% Homozygous polled (PP)     50% Heterozygous polled (Pp)     25% Homozygous horned (pp)

Table 2. Probability of a Polled Bull being Homozygous Polled if no Horned Calves are Produced No. of Polled Calves from Horned Cows Probability of Bull being Homozygous Polled 2 75.00% 3 87.50% 4 93.75% 5 96.88% 6 98.44% 7 99.22% 8 99.61% 10 99.90% 12 99.98% 14 99.99%
There are additional genes that affect horn-like growth, scurs, on an animal’s head. Scurs are incompletely developed horns which are generally loose and movable beneath the skin, not attached to the skull. They range in size from small scab-like growths to occasionally almost as large as horns. Because the gene for scurs is transmitted separately it has no effect on the presence or absence of horns. Not all horned cattle carry the gene for scurs and not all polled cattle lack scur gene.
The gene for scurs is expressed differently from the gene for polledness/horns. The way the gene for scurs is expressed depends on the sex of the animal. In males the scur gene is dominant, meaning that if only one of the two genes is for scurs the bull will be scurred. Therefore, it is easy to detect the scur gene in the bull and eliminate it from the herd. In females the scur gene is recessive, meaning that she must posses both genes for scurs in order for the cow to be scurred. If the cow possesses only one scur gene she will not have scurs herself but has a 50 percent chance of passing the scur gene on to her calf. The smooth polled cow may have the recessive scur gene, resulting in much more difficulty in identifying/eliminating the scur gene from the herd. Table 3 depicts the scurred inheritance patterns. The presence of the scur gene is indicated by Sc and the absence of the scur gene by Sn. These gene patterns are for polled animals, as the horn growth will cover up any scurred condition if it exists.

Table 3. Scurred Inheritance Patterns Genetic Makeup of Animal Cows Bulls ScScPP Scurred Polled Scurred Polled ScSnPP Smooth Polled Scurred Polled SnSnPP Smooth Polled Smooth Polled
Another factor that complicates the inheritance of polledness/horns is that in cattle with Zebu ancestry, like Brahman, Santa Gertrudis and others there is an additional gene that affects the inheritance of horns. Inheritance of horns in Zebu-type cattle is different from that observed in the British breeds. The polled gene (P), and the scur gene (Sc) can both be present in American cattle with Zebu ancestry. However, another gene, the African horn gene (Af) also affects inheritance of horns in these animals. The absence of this gene is expressed by the symbol (An).
Geneticists are reasonably certain that the way the Af gene is expressed is dependent on the sex of the animal, much like the way scurs are expressed. In males the Af gene is dominant to the polled gene, An. This means that a single Af gene will result in a bull being horned, even if he is heterozygous or homozygous polled. In females the Af gene is recessive to the polled gene An. In heterozygous polled females two of the Af genes must be present for the animal to have horns.
In animals possessing the Af gene in addition to the polled gene (homozygous or heterozygous) the inheritance patters shown in Table 4 can be expected.
Like scurs, the presence of the African horn gene is easy to detect in males since the presence of only one Af gene results in the male having horns. Therefore, progeny testing for the Af gene in males is unnecessary. If the bull is polled he does not possess the Af gene. If he is horned when his genetic ancestry shows that he should be polled, the reason may be that he possesses an Af gene. Cattle producers should also keep in mind that a proven homozygous bull will produce some horned calves if he is bred to horned or polled cows that carry the African horn gene.

Table 4. African Horn Gene Inheritance Patterns Genetic Makeup of Animal Cows Bulls AfAfPP Horned Horned AfAnPP Polled Horned AnAnPP Polled Polled
As you can see there are ultimately three pair of genes that may determine if cattle have horn-like tissue on their head in the form of horns or scurs.

——————————————————————————–Animal Husbandry Newsletter August 1996Department of Animal Science, North Carolina State University ——————————————————————————–Published by North Carolina Cooperative Extension Service, North Carolina State University, Raleigh, North Carolina ——————————————————————————–Distributed in furtherance of the Acts of Congress of May 8 and June 30, 1914. Employment and program opportunities are offered to all people regardless of race, color, national origin, sex, age, or disability. North Carolina State University, North Carolina A&T State University, U.S. Department of Agriculture, and local governments cooperating.