Originally Posted by RedsManRick
Say you're rolling a die and you've gone 30 throws without throwing a 6. You know that since there's a 1/6 chance of throwing a 6 on any given roll. Thus, in that time period you expected to have seen 5 6's, on average. So you surmise that you're "due" and expect a run of 6's sometime in the near future to "balance things out"... right?
This is called the gambler's fallacy
. Probability simply doesn't work that way. In reality, what's likely to happen is that from this point forward, you'll see 1/6 of your rolls showing up as 6's.
So, if Willy Taveras is truly a .330 OBP hitter, we would expect him to put up a .330 OBP moving forward, regardless of what he's done in the last few weeks. "Due" is true, but only to the extent that we should expect a player to perform as his true level of ability moving forward rather than to continue to slump. There should be no expectation of a hot streak to balance things out.
The mistake is made because we forget our assumptions. The most likely result of future events is based on the player's skill level. The player's skill level doesn't change because of what he did yesterday, or last month. Taveras won't be a better hitter tomorrow because he struggled yesterday. So if we want to know what is likely to happen moving forward, if he's the same player he was on April 1st, we expect a .330 OBP moving forward. If want to know what to expect by the end of year, we simply take the sample we've observed and add it to our best guess about what's likely to happen moving forward.
So, given that we know that his OBP through 218 PA is .294, and we would predict a .330 OBP over the remainder of his PA (say 350), we would expect his final OBP to be around .316.
Re-read what I said. The key is that I am talking about moving his career number back to his true skill level, .330, not having this season's number end at .330.
The difference between a players stats and the odds of a coin flip, is that the odds of the coin flip are taking into account every coin flip in the history of time, as you pointed out. The odds of Taveras getting on base is based only on his PA's, which are finite, and have a beginning and an end. That changes the effect the odds have on his remaining PA's, if we are too assume that the .330 number is fixed and reflects his true skill level.
Let's say that this is the only coin in the world, and we are going to flip it 1000 times. And the odds are fixed, so that it over the 1000 times, it will end up heads 500 times and tails 500 times. If the first 500 are 300 heads and 200 tails, the next 500 would have to be 200 heads and 300 tails.
That is exactly what you are saying when you are saying that Taveras' skill level is .330. You are saying that at the end of his career, he will have an .330 OBP. Therefore, if his career numbers are at .327 right now, in order for him to reach .330 at the end of his career, he must hit better than .330 to get back to his true skill level. (And I am hoping that he doesn't have too many more PA's as a Red.)
Now all of this is predicated on believing that his numbers represent his true skill level, and that they are fixed, like the odds of a coin. I think no one in this discussion believes that. However, that was the premise from which I was working.