Drop rates for set / legendary items
Solution 1:
In patch 1.05 the drop rates were doubled and Monster Power added a new bonus.
Using the base numbers from the Blizzard forums:
Max Max
Non-inf Inferno
base MF 100% 200 300 400 475 500 600 625%
normal 0.0036% 0.0072 0.0108 0.0144 0.0180 0.0207 0.0216 0.0252 0.0261%
elite 0.0440% 0.0880 0.1320 0.1760 0.2200 0.2530 0.2640 0.3080 0.3190%
It is now trivial to get 100 MF with paragon levels and NV stacks, so I usually assume the 100MF line as the new baseline. With the 300% cap in place, bonuses from NV and bonuses from MP the new max MF is 475% in Normal, Nightmare and Hell (produced experimentally by hitting the cap in Act 1 Normal Monster Power 10). The shrine does not bump this to 500%. I am uncertain if the upper limit is 625% or 650%, as I cannot get 5 NV stacks in Monster Power 10, Inferno, but if it behaves like Normal then the max is 625%, calculated as max. 300% from equipment + paragon levels + 250% from MP 10 in Inferno + 75% from 5 NV stacks.
Empirically, running between 325 and 375 I've been seeing a legendary or set item about every 4 or 5 hours of game play, using Act 3, Monster Power 3. Most of them stink, but that's another discussion all together. I've also had sessions where I've popped 3 in a matter of an hour at MF 260% MP 3 Inferno. I also see a legendary about every full act run at max MF in Normal (sometimes none but usually 1 maybe 2).
Keep in mind that each monster kill is an independent event. The odds are not cumulative, so killing 50,000 white mobs at 475% MF does not guarantee you a legendary pop (though I'd be somewhat startled if you didn't get one). You still only have a small chance on each individual kill to see a legendary or set pop. People often misinterpret the probability of independent events to think that doing more of the same will increase their odds. The odds on each kills stays the same. What changes is the odds against seeing another consecutive failure. See this question and answer for a mathematical treatment.
So let's look at the odds of seeing a legendary or set pop after repeated mob kills.
So what is this graph saying? Looking at the 100% MF for normal mobs line, it says that after about 3,000 white mob kills there is a roughly 20% chance that you will have seen a legendary pop. On the Elite 625% MF line it says that there is roughly an 80% chance that you will have seen a legendary pop after 500 elite kills. It is very important to note that none of these values ever equal 100%, even at 32000 kills the number is still "very close" to 100% but not quite.
Tracking these curves it is very clear that even for large numbers of kills the probability of seeing a legendary pop can be very low for normal mobs regardless of MF, but that it is much better to hunt elites for even less than 1,000 elite kills at any MF. It is for this reason that most serious legendary farmers are getting their 5 NV stacks and killing as many elite packs as they can find, not lingering about with white hordes.
So does this match up with the assertion that you kill 10-30 times the whites is as good as killing elites? Looking at the 60% probability line, at 100% MF against elites, if you are killing 10 times then that's better than the 100% MF against white mobs, it's acutally on par with 300% MF against whites. If you are killing 30 times then the 100% MF normal line is actually above the elite line. My battle net account shows that overall, including levelling at all difficulties I am averaging about 20 whites : 1 elite, just as an arbitrary number for consideration. As a level 60 I target elites, so I suspect this ratio is decreasing but it will take some time for all the levelling to be averaged out.
The maximum pop chance from a white mob at maximum MF is still about half of the base chance for an elite. This fits with my empirical experience. I have popped legendaries off white mobs but far more off concentrated attacks on elite packs.
One thing that changes the scenario somewhat, that I've failed to make note of in this answer is that an elite kill gives multiple pops many (not all) of which have a chance of being legendary. Normal kills only give 1 or 2 chances. As both NV and MP add extra items into kills. If we assume that an elite gives twice as many items as a normal (an easy assumption) then with my white:elite ratio we should only use the 10x comparison, and then clearly the elite killing strategy is modestly better. If we make assumptions about 5 NV stacks or higher MP levels then this gap widens.
It is very clear there are diminishing returns for adding MF. The distance between the lines shows there is a large gain for going from 100 to 300 MF, a smaller but decent gain between 300 and 475 MF but a very minuscule gain for pushing to 625%.
If there are hidden mechanics working behind the scenes or if the base rate is incorrect, then this picture will evolve with time.
Solution 2:
THIS ANSWER IS CURRENT AS OF PATCH 2.0.3
No solid numbers are available since the 2.0 patch and Reaper of Souls have come around, but what we have is a smattering of blue posts dealing with legendary drop rates and, in one case, a now deleted (but quoted repeatedly) blue post that sheds more light on how magic find affects drop rates in 2.0. It is unlikely that Blizzard will ever officially disclose their base rate for certain drops, so the best we can say is as follows:
The blue posts confirm that the chance has been elevated as of 3/28/14, and can be elevated further by playing on the various levels of Torment difficulty.
Certain legendaries and sets can only be found in Torment I or higher and like other legendaries their drop rate is increased with each level of Torment.
The "Magic Find" stat that used to be a staple on items in vanilla D3 have been totally removed from any new blue or yellow items in 2.0. There are still some legendaries that have this affix, but they are very limited. You can also still use a Topaz gem in your helmet to increase your chances, but the effect on rare and legendary items has been extremely nerfed. If you want to know how magic find is treated in 2.0, and how it is vastly different from vanilla, see my answer on the question "How does Magic Find work?".