What are the chances that you are on plane that has an incident? ICAO Annex 13 defines an "incident" as


  An occurrence, other than an accident, associated with the operation of an aircraft which affects or could affect the safety of operation.


Is there good record keeping about incidents globally? Not like Aviation Global Incident Map which only really only lists accidents and some airport issues.  

For completeness, the same document defines an "accident" to be


  An occurrence associated with the operation of an aircraft which takes place between the time any person boards the aircraft with the intention of flight until such time as all such persons have disembarked, in which (a) a person is fatally or seriously injured [...], (b) the aircraft sustains damage or structural failure [...] or (c) the aircraft is missing or is completely inaccessible.


Only incidents that an airliner pilot is aware mid-flight and can do something about -this a start to follow-up.  question. Exact likelihood is not important- is it 10% or 20%. For passengers, the likelihood is not important since pilots handle mainly well.
I don't know if there is a global compilation. But for Australian operations data exists to allow us to establish a likelihood of being in an incident. Accidents are so rare that it's difficult to get a reliable indicator, however if we are just measuring reported incidents I believe we could get somewhere.  

The Australian Transport Safety Bureau (ATSB) has published an "Aviation Occurrence Statistics" report covering 2004-2013.

Let's focus on high-capacity RPT operations, because lets face it, that is probably what most people are interested in. 

According to the report, in 2013, there were 597,000 departures of Australian-registered high capacity RPT or charter flights (unfortunately the report doesn't seem to split it into RPT only, possibly because we have a lot of high-capacity charter flights to the mines). In the same year there were 3292 reported incidents to the ATSB and 2 accidents. Therefore I would conclude that if in 2013 you boarded an Australian-registered high-capacity RPT or charter flight, your chances of being in a reported incident were around 0.5%, and chances of being are an accident is 0.06%. The incident percentage is approximately the same for the couple of years prior, so I am confident that this remains around about the mark today - though I would not consider the accident rate to be reliable because there were so few.  

The incidents cited in this data vary from loss of separation to a pilot being sick to birdstrikes. 

Have a read of the report but before you take it as gospel, keep in mind, as they say, "there are lies, damn lies, and statistics!"