> Why is it not something simpler, like P(bomb exploding in a year) ^ 100 ?

That would be "bomb exploding EVERY year (= this year AND next year AND ...)".

What we actually want is "bomb exploding this year OR bomb exploding next year OR bomb exploding this year and next year OR bomb exploding in two years OR ...". But that's too complicated, so we use the trick with 1 - P(bomb not exploding in a century).