Introducing (or perhaps re-introducing plagiaristically!) the Aggregate Games Behind Wild Card Second Spot stat! AGBWC2!
This tells you how desperate your dreams to be in the play-in game are. If you are a long way behind the team currently holding the slot, AGBWC2 goes up. If there are a lot of teams in front of you, AGBWC2 goes up. And if most of the teams are closer to the current slot holder than they are to you, AGBWC2 goes up too.
AGBWC2 (from now on, just AGB, since after all we're Royals fans with small dreams) is defined as the sum of all the games behind you are of all the teams in front of you. So say the standings look like this:
Alpha holding the WC2 slot;
Beta one game behind Alpha;
Royals 4 games behind Alpha.
Then AGB for the Royals is 4 (games behind Alpha) + 3 (games behind Beta) = 7.
The "3" above can't just be read off the current wild card standings; you have to subtract each intermediate teams' GB from how far the Royals are behind. For example, if there are five teams all tied for the WC2 slot, and you're three games behind each of them, your AGB is 3 * 5 = 15. If you are immediately behind the slot-holder, then AGB = GB.
The current AGB for the Royals is 2 + 1.5 + 1 (NYY,BAL,CLE, respectively) + 4.5 (the Royals behind Tampa) = 9.
This is actually one of the best AGBs the Royals have had in a while, but there's not much season left. The proper scaled metric would be dividing the AGB by the square root of the games left (assuming random walks so the variance goes as root(n)). The closest the Royals got in recent months was on August 12: OAK had the WC2 slot, and we were barely behind TAM and BAL, for an AGB of 11.5, with 66 games to play, for AGB/root(G) = 1.69. That value today is about 1.8, because of the few games left.
To really get a probability, the games behind each team would basically be multiplied together (since the odds of a biased random walk overtaking m other random walkers would decrease multiplicatively with m). It's actually possible to roughly do this; you get a convolution of something called incomplete beta functions, which are really not friendly creatures.
But the AGB above is good enough for just keeping track of the Royals' progress. Nine is pretty good; but it's pretty late.