How do I go about

writing a mathmatical prediction of how often a baseball game should end in a walk-off?

Maybe someone already did it.  Did you search up words:  probability baseball walk-off?

Walk-off home run or just a walk-off?

If you just mean walk-off then these statistics will give you a start: http://www.baseball-reference.com/play-index/inning_summary.cgi#gotresults&year=2011&team_id=ANY&ajax=1&submitter=1

Take a look at the last table. To get a starting frequency sum the number of games when a team that starts inning 9 either ahead or tied and ends behind and the number of games that went past 9 innings then divide this sum by two since about half of these games will have been won by a team scoring more points in the top of the inning then the opponent does in the bottom half. After that, divide by the total number of games.

I'm getting that roughly 4.6% (seems reasonable) of games ended in walk-offs in 2011. You could do this for more years to get a better estimate.