Why is the probability of likelihood of a sub-population more important than the likelihood of the population as a whole? I'm trying to find an easy lay answer to a specific popular paradigm regarding likelihoods in a population, or "more likely" in a sub-population over another sub-population.
Say you have $100$ cattle. $20$ of them are Holsteins, the rest are Longhorns. At any given time 20% of the Holsteins need the services of a Vet and only 10% of the Longhorns.
Now, most lay people might stop right here and say that the Holsteins are "more likely" to need the services of a Vet, since the percentages are so apparently different. They might even say that they are "twice as likely to need the services of a vet" as Longhorns do.
My understanding of probability is that given the population, the likelihood of something happening is out of the population as a whole. Meaning, if I were to pick any single cattle to find if they are in need of a Vet or not, it would be more likely that I would find a Longhorn than I would a Holstein. In fact, it would be twice as likely that it would be a Longhorn than a Holstein. Since, as a measure of the population as a whole, 8% of the population are Longhorns in need of a Vet, and 4% of the population are Holsteins in need of a Vet.
Am I wrong here?
If I am wrong, thank you. If not, what is the importance or value of using the sub-population percentages outside of the whole when talking about probability? I'm looking for my own disconnect in statistical math that makes this 20% vs 10% comparison valuable.
 A: now i may have misunderstood you but its seem you are confusing a couple of different question, or rather your definition of "more likely" is not well define. in your example of the 100 cattle we have 4 of the 20 Holsteins and 8 of the 80 Longhorns at the vet. now we can ask all kind of question about it. for example we can ask what is the chance of picking a Holsteins that need a vet (assuming we pick randomly), to which the answer is indeed 4% as we have 4 out of 100 to pick one, and indeed to probability to pick Longhorn that need service is 8 out of 100 which is double. but we can also what is the probability to pick a sick Holsteins or Longhorns, in which case the probability will be 20% and 10% respectively so the question is when someone say "Holsteins more likely to need the services of a Vet" what dose he mean? dose he mean that if i will go to the vet i will see more Holsteins than Longhorn?, clearly not we have 4 and 8, maybe they mean that if i will get a  Holsteins it is more likely to get sick the a Longhorn, in which case they will be right.
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now after this lengthy mumbling i would like to talk about what i believe (and of curse i may be wrong ) the source of the confusion, and is the assumption that there is a strong connection between the following questions. "given a sick cattle what is the chance it is a Holstein?". and "given a Holstein, what is the chance it is sick", people tend to thik that those switching of relation between what is given and what random dont chage the probability or that is a strong connection between the answer to those question, and while obviously if the likelihood of one is big it will affect the other in some way, one should be careful when takling about probability too not confuse the two as they are not the same!(the chance of a dodo to be dead is 100% the chance of something dead to be a dodo is not so high.)
