Post by eric on Jul 1, 2018 18:22:24 GMT
Let's say you have a baseball team of batters who do only two things: reach first safely or make an out.
Let's also say that the baserunners never take the extra base and that the fielders always get the lead runner.
We can then model any given inning like this...
...where in this case we are using a .500 OBP.
When the first batter gets up, there must be no outs and no one on base.
After the first batter, 50% of outcomes are no outs runner on first, the other 50% are one out no runners.
The second batter has a 50% of getting on regardless of what has already happened.
Thus we have 25% no outs runners on first and second, 25% one out runner on first from the first outcome of the first batter, and
25% one out runner on first, 25% two outs no runners from the second outcome, for a total of
25% no outs two runners, 50% one out one runner, 25% two outs no runner.
As this process continues our probability pools in three outs since there are only three outs, but
we score increasingly decreasing runs with each batter since there's always a chance we'd still be batting, and
eventually we reach an asymptote for runs in an inning relative to OBP, viz:
OBP runs
.100 0.001473333
.200 0.02328
.300 0.117334286
.400 0.37376
.500 0.9375
.600 2.06064
.700 4.27378
.800 9.09312
.900 24.01326The most obvious relationship is O^4. Let's divide through by that:
OBP factor
.100 14.73333333
.200 14.55
.300 14.48571429
.400 14.6
.500 15
.600 15.9
.700 17.8
.800 22.2
.900 36.6Yeah baby, yeah. I have an HUGE erection right now.
It's fascinating to me that this is manifestly a second order polynomial as opposed to, you know, linear. Or at least monotonic. We're not animals.
Now you can't quite see it with this arbitrary level of precision, but .300 ends in 142857 142857 142857. How do we get a seven out of .300? Well, 1 - .300 gives it, so let's multiply everything through by 1 - OBP:
OBP factor
.100 13.26
.200 11.64
.300 10.14
.400 8.76
.500 7.5
.600 6.36
.700 5.34
.800 4.44
.900 3.66Which brings us to this handsome fellow:
Say goodnight, Skynet. Now it's just a question of algebra:
(6x^2-18x+15)/(1-x)*x^4
3*(2x^2-6x+5)/(1-x)*x^4
polynomial long division time!
-x+1 goes into 2x^2-6x+5 at -2x, leaving -4x + 5
-x+1 goes into -4x +5 at 4, leaving 1
leaving 1/(1-x)
3*(4-2x+1/(1-x))*x^4And as a sanity check, when OBP = 0 we get
3 * (4 - 0 + 1/1) * 0
= 3 * 5 * 0
= 0
That is, if nobody ever gets on nobody scores.
and when OBP = 1 we get
3 * (4 - 2 + 1 / 0) * 1
= 3 * (2 + infinity)
= 3 * infinity
= infinity
That is, if nobody ever gets out everybody scores.
Everywhere.
Forever.
.
The upshot is that runs scored is very sensitive to OBP but in practical terms only when you get above .400. At .400 you're looking at three runs in a seven inning game, at .500 seven, at .600 fourteen.
