I asked in the off-topic forums how likely drawing only two techs in the modern age was
here. Sanabas figured out the probabilities (and his reasoning seems spot on). I had a .0461 probability of getting 2 techs as I did, and you have actually a bit over .6 probability of 7 tribes getting all 4 techs in the modern age.
For 7 tribes and 3 available techs, there are 3^7=2187 permutations.
The number of ways all 7 get exactly the same tech is 3, one for tech M, one for tech F, and one for tech E. So, the probability of 7 tribes getting one tech upon gifting equals 3/2187=.00137.
For all 7 to get exactly 2 techs we have say 1 M and 6 F's (7C1=7), 2 M's and 5 F's (7C2=21), 3 M's and 4 F's (7C3=35), 4 M's and 3 F's (7C4=35), 5 M's and 2 F's (7C5=21), 6 M's and 1 F (7C6=1). So, we have 126 ways to get exactly 2 techs. We have 3 combinations of techs. Specifically they are {M, F}, {M, E}, {F, E}. So, we have 126*3=378 ways that the AIs drew exactly two techs. So, we have a 378/3187=.1728 probability of 7 tribes getting exactly 2 techs.
Since 378+3=381, and 2187-381=1806, and either the 7 tribes draw exactly only tech, or exactly two techs, or exactly 3 techs collectively, we have a probability of 1806/2187=.8258
7 tribes getting exactly three techs at the middle age change, or the industrial age change.
If we consider all three era change events, no one of them influences each other... at least so far as any of us know. So, we can treat them as independent events and apply the multiplication rule for probabilities. It follows that, the probability of getting 3 middle age techs,
and 3 industrial techs,
and 4 modern age techs from 7 scientific pals equals
(9912/16384)*(1806/2187)*(1806/2187)=.41255. That's much higher than I expected.