I don't know whether to post here or r/askscience, but I'll start here.
The story of Apollo 13 is mesmerizing, but behind all the exaggeration and fable how unlikely was the crews survival after the catastrophe on that mission?
Was their survival thanks to the actions of the crew aboard and NASA (like the creation of the unorthodox air scrubber), or were there any instances of pure luck or barely controllable variables in which the dice were rolled in their favour? Also, if all the problems could have been handled without luck but by rational thought and the tools available, how short were the time constraints during these problems and did they serve as a very real chance of death?
Speaking here not as a Professional historian but as an Aerospace Engineer; there was quite a bit of luck involved, but quite a bit of skill too. Using the lunar rover descent stage module for an orbital correction burn was completely outside the scope of the design of it, but the Grumman engineers ran the numbers and gave it the thumbs up. The astronauts had to do a controlled burn with no trajectory / guidance computer because they had to power down the command module to save enough juice for re entry. They did this orbital correction burn by watching Gymbals and trying to keep the earth in the window while firing the motor manually with a stopwatch. This took a lot of skill. It's extremely hard to orient in space as attitude control is a huge issue, so keeping it stable is hard enough, but burn for even half a second too much or too little and their reentry angle is too steep and they burn up or shallow and they ricochet off out into space with no hope of recovery. Luck plays into factor with regards to exactly what was damaged. Was the reentry shield, an ablative coating that evaporated heat away from the craft damaged? Will the craft survive reentry? Will there be enough voltage to power on the guidance computer AND deploy the chutes? They just didn't know. They also never ran a situation test for a full power down of the command module during flight. Would condensation on the panels short things out? They were down to milliamps in spare power, they couldn't lose anything.