An alien contra-organization wants to spur the development of Earth to help humans in the near-future in presence of a prime-directive like law. They cannot establish actual contact, so they do the next best thing: covertly deliver some important technology.
Unfortunately they do something not-so-clever here. Assume our secret organization replaces some percentage of batteries P with new, upgraded batteries that do the following:
- About 90% of the mass of the new battery is antimatter (the technology is very miniaturized, so most of the 10% is the outer shell for point 3).
- It converts this into energy using the matter in air at near-100% efficiency.
- It looks exactly the same as the old battery from the outside, and performs the same, with the caveat that:
- The new battery never seems to run dry...
The replaced batteries are completely random, and could be anywhere. In multi-cell batteries, each cell has its own probability P of being swapped. We also assume the replacement happens at-once. Exactly how that is done is left unanswered and isn't the point of the question.
A problem arises when somehow one of the new batteries is destroyed. The result is a large thermonuclear explosion (minus the fallout). A standard AA battery would produce about a 1.25 MT-equivalent, a 9V battery would produce a 1.87 MT blast. For the P=1 case: a Tesla pack weighs 540 kg. that's about 24 GT.
If we plot the human survival rate R against the probability P, we know two points: at P < 10^-10, R = 1, as it's likely that nothing happens. At P = 1, R ~ 0. It's likely the only survivors are in very isolated places. Say: in space, aboard a submarine, aboard a ship that isn't in the very busy shipping lanes ,rainforest natives, or the antarctic as any battery that explodes sets off a giant chain reaction.
Our organization wants to know, what would be an optimal P? Thus, they want to know: what would the graph look like in-between these two P values?