It's April, 1945 and you're a high ranking scientist in the Manhattan Project. You also happen to have the ear of President Truman, so your words and actions have a major impact on world outcomes. Everyone has been working hard on the bomb, determined to defeat the Nazis, except that the Nazis are presently being defeated and you're not nearly as concerned about the war in the Pacific. (You know Japan will lose eventually, regardless of the bomb.)
Another scientist points out to you a disturbing concept: It is possible that the forthcoming A-bomb test (scheduled for July) will cause the atmosphere itself to ignite in a self-sustaining fusion reaction that would consume the entire globe. The consequences would presumably be the extinction of the entire human race and most life on Earth. You don't know if this could really happen, so you do some back of the envelope Fermi estimates and you at least can't disprove the possibility. You might think it's a 0.01%, 1%, 10%, 50%, 90% or near-100% chance. [Pick whichever makes the thought experiment most interesting to you.]
How do you react? Do you resign? What would you tell the president in your next conversation with him? Do you advocate that nuclear testing and the Manhattan Project be immediately halted? If it is halted and the war ends without a test being done, do you advocate for an international ban on testing until the question of atmospheric ignition can be resolved? And how would you propose to resolve the question, from a physical and engineering perspective? (Maybe we can test fusion temperatures inside a device like a particle accelerator or tokamac. The tokamac would be invented a few years later, so you could probably conceptualize it. Or maybe an atomic test could be done deep underground or on the moon.)
Given the stakes, none of the probabilities you mention have enough nines to make me comfortable. I would recommend doing basically the same thing in all cases, which is to produce a decent argument and convince a bunch of physicists, then convince the military leadership of the manhattan project, then convince the political leadership.
Experimentally, my wrong-physics-subfield guess is you're going to want to do things like measure interaction cross-sections of nitrogen nuclei. You can do this with a room-sized particle accelerator, I think.
Huh, interesting. Definitely an example of why multiple semi-independent lines of evidence is a good idea. I wonder if you could get the relative rates of hydrogen and nitrogen fusion out of the distribution of elements in the sun, even without having to know its age... except of course we're made out of leftovers, which means you can only put a bound on it.