This just in, Eye Chart's play on Kearns' grounder in the 7th made the #2 Web Gem tonight, behind a diving stop and throw while literally sitting on his ass by Omar Vizquel. Too bad Vizquel had a low throw an inning later on a potential double play with the bases loaded, which turned into an error, allowing 2 runs to score.

Do I have something backwards (wouldn't be the first time)?

But if King's GB/FB ratio is lower than normal, doesn't that mean we might expect it to trend back up over a larger sample? Meaning, more GB per FB will be seen upcoming?

And I interpret that as a good thing.

Unless there's but another corner of sabermetric thought I have yet to explore or I got my acronyms goofed up.......

Does anyone remember when Johan Santana was in the bullpen, to save a 5th spot in the rotation for (drum roll) Kenny Rogers (who is still a MLB pitcher and doing pretty darn well)? At the time, the Twins also had Guardado and Hawkins in the pen, making them a wicked force. They needed it, with little offense or SP, but it was an interestingly effective strategy for a whole year.

I guess it depends whether you're an optimist or a pessimist Ryan. In the latter case, you might fear that this is some sort of sign of decline for King. Obviously, with the small sample size and all, you'd have to be a pretty extreme pessimist though.

Basically, I just noticed that in 2 of his last 3 appeearances, King was visibly upset about being pulled from the game, so I wondered if there was something Tony knew that we didn't, and that tidbit about the GB/FB ratio was the best I could come up with.

King hasn't even thrown a total of 8 innings yet this year, so I wouldn't read too much into any numbers. Without checking, my recollection is that he got of to a relatively slow start last year, as well. A little more woriisome to me is that he hasn't seemed to have the command he had last year, but like I said, he's only thrown 7+ innings so fat this year...

Regarding the bullpen issue, is there anyone we could reasonably trade for?

I think our bullpen is in fine shape. Tavarez has been solid of late, I think Flores and Reyes are a great lefty-righty tandem, I think King will be fine. That's four decent relievers that I trust to come into a game--nbot many teams have that kind of depth. Cali and Journell haven't shown much yet, but they're stopgaps and will either improve or be gone once Izzy, Lincoln, etc. are back.

What is the scoop with Lincoln? He looked great in limited action last year (high ERA was the result, I thin, of one bad outing and/or terrible BP support) and I know he's on the DL, but is he expected to be a contributor any time soon?

I fear "Slider Mike" has had more setbacks than Michael Jackson lately... I get the sense that the Cardinals are going to wait until Lincoln is actually able to go on a rehab assignment before considering whether he can pitch again in the bigs.

Ryan:

We can expect Kings GB/FB ration to rise, but only to his career level (actually to his level of ability, but we're assuming that his career level is his level of ability. A plausible assumption). His GB/FB ratio won't rise higher than that just to get his yearly numbers back in synch with his career (unless he starts to suck).

Think of it this way: If Player A is a .300 hitter but hits .200 for the first month of the season we could expect them to hit .300 for the rest of the sason leaving their seasing average at ~.290. We wouldn't expect them to hit .310 to bring their season average up to .300 (Assuming their natural skill level is '.300 hitter').

All of that holds true if the deviation from the norm is just random chance, it could be the case though that King has some sort of problem that is causing his GB/FB ratio to be out of whack.

Yeah, I suppose that's one way to look at it. But if you're assuming King's "Prior" GB/FB is a reflection of his actual ability, which I'm assuming hasn't changed (I see John's point, that it may in fact have changed, causing the drop, yikes), then it's also natural to think of his average GB/FB as the mean in his sampled distribution, in which case, we expect fluctuations in any sample and in any direction from his mean.

If you flip a true coin a bunch of times, and get an unnaturally long string of Heads early on, you can keep flipping it, and eventually you'll see some combination of more Tails to balance it out. The true mean doesn't change (in King's case, it may have), so you *would* expect a higher than normal off-setting action... eventually. You just can't predict when. That's the gambler's fallacy.

That's why in one of my fantasy leagues, I picked up MIke Lowell, who's batting less than .200 now I think, and who is a really good hitter normally. Yet he's also getting older. Am I witnessing a long bad streak signalling his decline or am I about to reap the benefits of his upswing? (I'll let you know by the end of the season).

But you're right in your example, I wouldn't expect some hitter to go hit .400 the second month to bring his average back up so tidily. But if he was truly a .300 hitter, I'd expect to see some months above .300, some below, regardless of his first month.

However you interpret it, it seems his GB/FB ratio should go up to more near his career numbers. Whether it balances out for the year or not, it'll be an improvement.

Er, and I assume groundballs are still favorable to fly balls!

>If you flip a true coin a bunch of times, and get
> an unnaturally long string of Heads early on, you
>can keep flipping it, and eventually you'll see
> some combination of more Tails to balance it out.

No. This is a common misconception. Each time you flip the coin you have a 50% chance of either outcome. The coin has no memory of having come up heads and needing to come up tails now.

You're right you would expect a hitter to fluctuate around his true ability. But you wouldn't expect him to fluctuate around some new number that would result in a .300 average.

Well, I guess you're talking about the iid assumptions.

But what assures the coin flips are indepenent of each other and unpredictable (beyond their bias) is that they come from the same hidden distribution, and that distribution is fixed (unlike ball players).

If that distribution is fixed, and each coin flip is independent, then what I stated is true, given a large enough sample, since you will by the definition approach the hidden fixed distribution through the limit.

Meaning, if you flip a true coin and you get 10 heads right off the bat, you will eventually even this out so that the larger sample average approaches p=(.5). This could not happen if you did not see more tails as often as you saw more heads. And i'm not saying tails has a greater chance of coming up at any given flip, you're right about that. I'm saying to actually ensure that property, you must see more tails eventually. It's a subtle sort of paradox I know.

Ryan: I'm sorry but you are completely wrong.

I flip a coin 10 times and get 10 heads. If I flip a coin 10,000 times and get 5k heads and 5k tails then the average h/t goes from ~ (or 1) to .5, not because i've flipped tails more but because the larger dataset obscures the small anamoly in the beginning.

If I flip the coin 5million times the 10 heads is nothing.

Each coin flip is independent.

Some reading:
http://en.wikipedia.org/wiki/Law_of_averages
http://arnoldkling.com/apstats/coins.html

Josh, I'm making a subtle point about probability theory which I may not be making well.

To illustrate using your own example:

In your own 10k=n example with 5k heads and 5k tails, simply remove the first 10 heads from the sample. now the n=9990. What's the ratio now? See? By removing the initial 10 heads from the sample, you have created a new sample with 10 "extra" tails. So you say "not because i've flipped tails more" but in your example, *you have* flipped tails more!

I'm not saying that the observation of the first 10 heads gives you anymore predictive power for any flip or series of flips other than p(heads)=.5. However, to preserve the infinite limit, you, by definition, must see a compensating number of tails or you may challenge the assertion that the coin is "fair", that p(heads) = .5. There are other ways to state this situation to make the point as well, but it would get even wordier.

The supporting reading I could give you for this is not online to my knowledge. But if you want to buy a book about it, check out "An Introduction to Probability Theory and Its Applications" by William Feller, part of the Wiley Series in Probability and Mathematical Statistics. It is probably part of any university math library as well.

This debate we're having is one of the reasons I study probability. On some levels it is completely obvious, and at other levels it seems to contradict not only intuition but itself. Cool stuff.

Ryan: Sure if you take a sample and remove 10 events of one type (the first 10 heads) the remaining events will be weighted towards the one you left in. Thats not probability that's just mucking with your data set to get results you want.

But if you flip 10 coins and they all come up heads that has no bearing on the next 9990 coin flips. Those will distribute 50/50.

You seem to be claiming that there is some hidden value of probability that can assert it's will on events....

I have to go, but more on the limit problem later, but basically as x approaches infinity the number of successfull trials isn't adjusted to keep balance, but the number of unprobable events becomes numerically insignifigant.

back.

Your example is incorrect because it uses a specific data set and draws conclusions from it. What if my 10,000 flips resulted in 5,060 heads and 4,940 tails. What if the first 10 flips were heads? By your logic the following 9990 flips should have leaned towards tails slightly to make up the difference. But that result is still perfectly valid from a probability standpoint. So why didn't the magic force intervene to maintain the balance?

Any small sample is likely going to be further away from the 'true' outcome than a large sample, because numerous iterations will tend towards the true percentage outcome (As x -> infinity the ratio will converge on the true outcome). However the large sample wasn't 'correcting' the small sample it's just random chance, and it's completely possible that the small sample will be off in the same direction as the larger sample (since for values of x < infinity the ratio is unlikely to exactly match true, and may be off in either direction).

To recap a low probability event (10 concurrent head flips) does not mean another low probability event will occur (a smoothing affect to meat the 'true' ratio). It is possible that this will occur but it is the result of random chance.

Oh man. After two softball games and a few jugs of beer...

Yeah, I think you basically summed up the main points of prob theory better than I did.

There are a couple smoothing points I might make:

"To recap a low probability event (10 concurrent head flips)..."

10 heads in a row is just as likely as 10 tails or any other sequence(just 1/2 to the 10th power), so it's not really a low probability event (anymore than is drawing a 2 rather than a K from a deck of cards), unless you measure it in a different way (like sums of all 10 flip sequences with heads = 1, tails = 0, in which case 10-sequence sums of 0 are equally likely as 10-sequence sums of 10 and much less likely than 10seq sums of 5 -- like rolling 7's in craps compared to 2).

In infinite sequences even a sequence of something like 10,000 heads in a row is not a low probability event either, since it is guaranteed to happen and you can compute how often, on average, you would expect to see such an event as well as that variance through time (trials), which is sort of ridiculously large. In the same sense I mentioned above, it could be considered a low probability event, but again that's a different measure.

"But if you flip 10 coins and they all come up heads that has no bearing on the next 9990 coin flips. Those will distribute 50/50."

It's true that the first 10 coin flips have no bearing on the next *you name it* number of coin flips, and by that same token you don't know that the next number of coin flips will distribute 50/50, they will more than likely deviate from that because

"You seem to be claiming that there is some hidden value of probability that can assert it's will on events...."

there is a "hidden value" which is defined by the limit

"(As x -> infinity the ratio will converge on the true outcome)"

that puts p(heads) = .5 (this was my original point, that the "magic force" you speak of is not really by magic but by definition and only applies to the infite limit, not any samples of it).

But if you take enough samples (who knows how much "enough" is?) you will get about as many with extra heads as with extra tails in it, and even then the ratio is not likely to be exact. It is still not possible to predict which ones will have the extra tails, but the extra tails by definition have to show up. These were the elusive extra tails I was referring to.

I'm trying not to slice hairs with you, I think we're eating the same piece of cheesecake from opposite sides.

Wrap your heads around this, stat nerds: What if you had a coin that was heads on both sides? Woah...

Man, I could make some money with one of those coins.

John, you also might be interested in meeting with a friend of mine, who has this bridge. He's in a bit of a jam and needs some cash and *might* be willing to sell it to you...

Let me know.