In Mage the Ascension, coincidental magic is best described as a mage influencing reality to make something happen that might. It is easier to accomplish than vulgar magic by far because coincidental magic will fit the perceptions of Sleepers- and thus, does not require the mage to alter reality as much as vulgar magic does.

Example- a mage from one of the nine Traditions decides that she needs money for a personal project. There are a number of ways she could go about acquiring this magically- one, she could wave her hands and poof, a wad of twenties would appear in her hand. No matter where you are, though, this is vulgar magic- money doesn't grow on trees, nor does it appear in mid-air. If she wanted to be subtle, she could pull it out of her wallet, but it wouldn't be a whole lot... people don't carry five hundred dollars in their wallet, it just doesn't happen very often. If more money was needed, she could go purchase a lottery ticket and insure that it was a winner. While winning the lottery doesn't happen very often, it IS somewhat plausible... and thus, the magic remains coincidental.

What passes as coincidental magic is often dependant on the nature of reality. In the laboratories of the Technocracy, a Verbena mage cutting her palm and letting the blood drip on a mainframe with the intent of frying their computer system is going to have a lot harder time than a Virtual Adept hooking up his computer to the Technocrat system and uploading a massive virus. By the same coin, when one visits the Amazonian jungles, a Son of Ether or a Technocracy agent is going to find that their cybernetic GPS system is less efficient than a Verbena carving a rune into his arm and asking the world where they are.

The status of what is coincidental magic and what is vulgar is often a sign of how encroached one of the factions in the Ascension War is on the territory. The Technocracy owns most of the Western world; thus, their ideas are more efficient by far. When one visits a Tradition chantry, though, the ideals of magic they espose are more likely to work- however, Tradition mages are almost always forced to go to the Umbra to perform some of their stronger magical ideals. Marauders, by nature, are rather indifferent to what is coincidental and what is not, and the Nephandi will not use magic until they know their sweet lies have failed.