The first person to solve this problem will win a decent Uhkapian Serpent (5 million gold value), a nice wand (5 million gold value), and 1 million in gold.
Sam wants to craft a Hellfire ring. He has three Infernal Machines, and only needs one more organ to craft it.
This is telling you something important. The only drop rate that matters is the drop rate coming from the portal of the organ he needs. Lets say the organs are A,B,C. If he only needs one more organ, WLOG let him have organs A,B.
In MP5, the portal for C will drop C at 50%. He has three machines. HE HAS THREE MACHINES. THAT MEANS HE CAN OPEN EVERY PORTAL IN ONE GAME.
Boom - opens a portal. Is it the one for organ C? No? Then drop rate is completely irrelevant, he DOES NOT NEED IT. Who CARES if this organ (A or B) drops or doesn't. It is completely irrelevant.
Open second portal - is it organ C? Let's assume so. In mp5 it has a 50% drop rate. So the chance to get the organ is 50%.
Suppose the second portal wasn't organ C. Well whatever portal it is, it is not C nor the one before. Does it matter? NO. !@#$%^- IT, NO. He DOES NOT NEED IT. Kill the uber, carry on.
Finally the last portal is guarenteed portal C. 50% drop rate. Done.
The portal does not matter. He has 3 machines. He has entry to each portal in one game. Therefore the drop rate will be 50% for the organ he needs.
This is called the "memoryless" principle. What happened beforehand (which portal opened) is irrelevant when the portal is finally opened, BECAUSE he opens ALL THREE.
Edit: forgot to say therefore the chances are at least 50%. Obviously these aren't optimal. But for anyone who says 50%, no.