So, as far as I'm aware a well-trained, veteran, professional army has a significant advantage over an army which is made up primarily of levies or fresh recruits.
But I was wondering how much this advantage realistically is in numbers. All else being equal (so the same types of troops, not cavalry vs. infantry or something, no huge disparity in technology where one side has gatling guns and the other is using spears) historically how large an army of inexperienced soldiers can a professional army generally take on? Two times as many? Three times? Four times?
I'm primarily looking for specific historical examples of this happening (a smaller veteran and/or professional army beating a larger non-professional army or army of fresh recruits) and the respective army numbers involved. The larger the disparity in the example, the better. I'm especially interested in the extremes, though more average ones are also welcome.
Thank you in advance. =)
Edit: I just wanted to clarify that I know there is no perfect scenario here. There will always be other factors involved. Ideal examples are the ones that are as even in ways other than numbers and professionalism as possible, obviously. But I know that one which comes down solely to training and professionalism probably isn't around. Hopefully though by seeing many examples I can look at these examples all in closer detail, see what other factors played a role (which will tend to be different for different examples) and kind of try to sift them out that way and get a vague idea what the comparative advantage is.
It is impossible to offer anything like a historical answer to this question. You're trying to isolate two factors - training and numbers - to determine how they weigh up. But there are no historical examples in which these two factors operated in isolation. This completely spoils the scenario. At the most basic level, a trained marksman may be more effective with a rifle than someone who has never held one before, but a stray bullet will kill him just as dead as an aimed shot. As you zoom out from there, the individual ability and group training of a warrior rapidly lose even more of their significance compared to larger factors like morale, supply, battle and campaign planning, random happenstance, and the extent to which the military means available are suited to the intended ends. These things are specific to each military operation and they make it impossible to formulate any rule along the lines of "1 professional soldier is worth X untrained levies". The real question is always: to what extent can a professional army make its advantage count?
An example. Just before he invaded mainland Greece in 480 BC, Xerxes boasted that there were some among his professional Immortals who would easily be able to beat 3 Spartan levies at once (Herodotos 7.103.5). It seems a reasonable claim; the Spartan army was a militia which received no combat training and had little recent experience of war, while the Immortals were the only professional soldiers in the Mediterranean world at the time. At the decisive battle of Plataia, the 10,000 Immortals were only slightly outnumbered by a Greek formation of 11,500 hoplites (of which 5,000 were Spartiates) - surely no great challenge for these skilled and versatile Persian infantry. Yet they barely made a dent in the Greek battle line.
How is this possible? Herodotos himself stressed that the Spartans wore more armour, had longer spears, and held their formation despite the storm of arrows and the constant Persian attacks. We can also see that the terrain favoured them; neither side wanted to fight uphill, but the Persians were tempted to try it anyway because they thought the Greeks were already fleeing. Finally, a Spartan managed to kill the Persian general Mardonios with a lucky throw of a rock, and the sight broke the Immortals' spirit. None of this has much to do with the relative level of training or professionalism of the troops, and their numbers weren't that far apart. The battle was entirely decided by other factors negating the Persian advantage.
I'm sure you'll be able to find some examples of more professional and better organised forces holding off innumerable waves of enemies. But these will invariably be cases where a huge range of other processes - including the enemy's fighting methods and tactical plans - had already played out in favour of the professional force. This is never predetermined. While better training and organisation usually help reduce the risk of the kinds of panic and confusion that tend to be the root cause of defeat, they do not make individual fighters superior to untrained counterparts in all situations. Anything can happen at any moment to spoil the ability of even the best-trained army to hold together in battle. That means the advantage of such a force can never be expressed as a mathematical formula. For every battle of Watling Street (where the Romans won despite allegedly being outnumbered 10:1 by Boudicca's forces) you'll find a battle like Plataia, in which inexperienced recruits defeat an almost equal number of professionals by exploiting terrain, technology, intelligence, surprise, or simply by outlasting them in combat.
Indeed, in most cases where a small professional army triumphed over a large mass of levies, it probably happened because the commander of the larger army made mistakes - overcommitting too soon, trying to win through brute force, failing to restrain ill-disciplined forces, or allowing them to be goaded into an attack. The commander of the professional army will actively try to engineer such situations (like Mardonios at Plataia, probing the Greek line for days in the hope of tempting them off the high ground and into the plain). But they cannot count on these things to happen, and the assumption that they can just rely on their troops to be worth X number of their enemy would be extremely dangerous. There are few greater threats to armies than overconfidence; Xerxes paid dearly for his arrogant claim that the best of his Immortals were worth three Greeks in battle.