The numbers? Well, it's not simple.
Here's an "Expected Runs" calculator that I found online. You can read the author's explanations and form your own judgment as to how valid it is.
What it shows is that the likelihood of scoring one run, with a runner on second and nobody out, is 37.9%. The likelihood of scoring one run, with a runner on third and one out (the situation if the batter sacrifices sucessfully), is 48.8%. So this indicates that the bunt, if successful, increases the chances of scoring that winning run in the bottom of an extra inning with the score currently tied.
But that's not the whole story, of course. If the bunt is not successful, the likelihood of scoring that run...go down.
If the bunter strikes out or pops up, the odds drop to 25.1%.
If the bunt is too strong, and the runner is out at third (result: one out, runner on first), the odds of scoring drop to 12.6%.
If the bunt results in a double play, the odds drop to 4.5%.
And this one, which is surprising: If the bunt is successful, and the bunter not only advances the runner, but also reaches first base successfully (so there's nobody out, runners on first and third), the odds of scoring one run drop from 48.4% to 43.3%.
Now, this last result makes me wonder at the model being used...but it's based on a huge amount of data...and when I think about it, there's a multitude of possibilities for "getting out of the inning" that open up with that runner on first base (for example, strikeout followed by ground ball DP) that don't exist with just the runner on third.
Moreover, all these above probabilities are averages. In a given situation, the abilities of the specific batters that will be at the plate are going to significantly affect these odds. Is the batter a good bunter? Does he have speed to have a good chance of reaching first and avoiding an out? What about the batters that follow the bunter? You probably need a hit or a relatively deep fly ball to get that runner home; is the one-out batter likely to deliver that? Depending on the batters (the first and second hitters in the inning, in particular), the actual likelihood of getting the run may well be higher if the first guy swings away.
