Insider trade on Splunk acquisition?

I know this isn't what happened, but what if one day I'm waiting for the bus and I over hear a guy talking on their phone about an imminent acquisition?

1. Would that still fall under insider trading even if the information was accidentally heard, and even if I wasn't 100% sure of its accuracy?

2. If I had no clear connection to the company how would it be proven that I was trading on insider information? Surely it's not enough just to say the trade was statistically unlikely, or is it?

It's still not public information (e.g. "Material Nonpublic Information"). You can't trade on it.

See 17 CFR § 240.10b5-1 "Trading “on the basis of” material nonpublic information in insider trading cases", particularly section (b) "Awareness of material nonpublic information."

https://www.law.cornell.edu/cfr/text/17/240.10b5-1

No this is not correct. If someone spreads a rumor and you don’t know that the information is from an authoritative source it’s not MNPI. It’s speculation.

If someone anonymously posts on Twitter that Splunk is going to get acquired by Cisco and you immediately buy options, and you don’t know who this Twitter account is, then it’s speculation. Also it’s no longer non-public information if someone posts it on Twitter.