I have recently seen some debate in forums and FB groups, where some people say it is good idea and some say it is bad idea to have 4 people running 4 radiant relics at the same mission.
I am not a scientist, so don't take this post as the ultimate truth, but just something what my common sense is telling me.
Numbers on this page are based on what is written at http://warframe.wikia.com/wiki/Relic today and may change over time if DE changes something in the game in future patches.
Drop chances when using radiant relic are as follows:
Common - 50%
Uncommon - 40%
Rare - 10%
As there is 3 Common items - 16.7% per item
As there is 2 Uncommon items - 20% per item
As there is 1 Rare item - 10%
Wikia also says if 4 people all have radiant relics equipped and running them so called "parallel", then the drop chance of Rare item is 34%. How someone got this number - I don't know.
IMHO it is bad idea to run 4 radiant relics at same tie just because "what if" all 4 people draw Rare item from their relics? Mathematically that is not very likely, but it can happen sometimes just like sometimes people do win the lottery. In this case, it would mean that 3 rare items just went down the drain.
Sure, running the 4 relics one by one in team of 4 takes more time, but rewards IMHO are much better.
Lets do the simple math.
50% chance to get a Common item
40% chance to get a Uncommon item
10% chance to get a Rare item
Same chances for the 2nd 3rd and 4th run. So add those 4 together and you get:
200% chance to get a Common item
160% chance to get a Uncommon item
40% chance to get a Rare item
So based on the adding up the chances and user picking always the best rewards in 4 runs - the Rare part drop chance is 40% and for the Uncommon part 60%. The other 300% can be ignored, because those low rarity items are not picked in the end of a mission/wave (because You only can choose 1 out of 4 rewards in the selection screen).
This 40% Rare and 60% Uncommon IMHO means better odds than the Wikia mentioned "4 parallel rare relics" giving you a 34% chance for at least one rare item.
So in other words, you will most likely get 1 Rare part and 3 Uncommon parts out of 4 relic runs, 2 Rare / 2 Uncommon if you are lucky. If you get more than 2 Rare parts - you really should play The Lotto instead of warframe. :)