Interesting. Something's a bit odd, though. If the events are rare, then it's hard to know what the correlations are with any precision. If the events are common, then, yes, we should be able to see the anti-correlation, but this would be a really bad sign -- there'd be no reason to think that the disastrous event where both co-occur isn't right around the corner.

ETA: I exaggerate a bit. There'd be no reason if the independence model was true. If, in reality, there was some circumstance specially protecting us somehow the situation wouldn't have to be dire.