The "1/36 chance" and the "negligible" aren't the corresponding/analogous probabilities. 1/36 is P(detector "yes" | not nova). The "negligible" is P(nova).

What we want to calculate is P(nova | detector "yes"). Using Bayes' rule it's

P(nova | detector "yes") = P(detector "yes" | nova) P(nova) / (P(detector "yes" | nova) P(nova) + P(detector "yes" | not nova) P(not nova)) = 35/36 * P_nova / (35/36 * P_nova + 1/36 * (1-P_nova))

P(nova | detector "yes") > 0.5 if P_nova > 1/36.

This is also quite intuitive. Unless going nova is more likely than malfunctioning, you should bet on malfunctioning. This is very much like David Hume's quote:

"no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish." (test: would you say the same amount of "holy crap! WTF!!" if it turned out that the testimony was false, compared to if it turned out that the miraculous thing actually happened).

So: no detector "yes" is sufficient to establish the sun going nova, unless the detector saying "yes" be such, that it lying (1/36) would be more miraculous, than the sun going nova.

It's fascinating how precisely this can be formulated and how clear it is with probability theory.