The "once safe always safe" defies the rules of probability and basic logic.

So I turn on the TV with 5 minutes left and team X is ahead by 20, which is a safe lead according to your calculator. Team Y though makes a tremendous run and with 1 minute left the lead is one point.

And you're telling me it's still a safe lead? Right. What if it were tied, or team Y had pulled ahead by 2? Would team X still have a "safe" lead when team Y is leading?

This is of course based on a basic logical fallacy and ignorance of probability. If I toss a coin and it comes up heads 10 times in a row, the probability of it coming up heads on the eleventh toss is still 50%.

Believe it or not, little angels do not come down and alter the laws of physics so that the coin is more likely to land tails. And God is not going to come down from heaven and play D for team X to assure they win just because they lead by a certain amount at some point in the past.

I'm afraid you're incorrectly criticizing the article and misunderstanding probability here. You are confusing probability at a given point with conditional probability at a different point.

First, let's get one thing clear. Bill is not defining "Safe" here as absolutely safe in a literal meaning, he's using it to mean that overcoming the odds would be astoundingly improbable. Exactly how improbable that would be was not really defined, but let's define it as less then 1 game in all the NCAA sanctioned games ever played (which does not mean that it hasn't happened).

Second, let's use a simpler example, say your coin flips. Let's look at the odds of getting 10 heads in a row. The absolute odds of that would be 0.5^10~=0.1% or about 1 in 1000. Not small enough for the definition above, but it'll work here.

Now assume that we start flipping, and you get 5 heads in a row! So now the probability that you will get a total of 10 heads in a row is really reduced to getting another 5 heads in a row. So now your probability of success is 0.5^5 ~=3% or about 3 in 100. That's 30x more likely!

However, that result is a conditional probability. It's the probability of getting 10 heads in a row given that you've already gotten 5 heads in a row. What your post claims is that because the conditional probability is much higher, that the original probability is too low. But it's not. The conditional probability does not affect the original probability.

Going back to Basketball, this means that Bill's assertion still holds, the chances of a team coming back from 20 points down with 5 minutes left is still vanishingly small, even if later they might close the score to 2 points with 30 seconds left. That is not the same thing as saying that their chances of coming back from 20 points down with 5 minutes left given that they've come back within 2 with 30 seconds left is vanishingly small (which is obviously not true). At that point they've already got their 9 heads coin flips in a row (the improbable/difficult part), getting the last one is relatively likely/easy.

Bill James admits this is a heuristic model. While there is one instance where his tool does not hold true (the Duke-NC State game), in every other real world situation it has held true.

In the article James freely admits that his safe lead calculation does not rule out the possibility that a team down 30 with 10 minutes left may go on a furious run, he just says that the combination of things that need to go right to do this is so immense that it has only happened once before. So, for all practical purposes, the game is over when his tool says it is.

"Once a lead is safe, it's permanently safe, even if the score tightens up. You're down 17 with three to play; you can make a little run, maybe cut it to 8 with 1:41 to play. The lead, if it was once safe, remains safe. The theory of a safe lead is that to overcome it requires a series of events so improbable as to be essentially impossible. If the "dead" team pulls back over the safety line, that just means that they got some part of the impossible sequence—not that they have a meaningful chance to run the whole thing."

This pretty clearly confirms the OP's point; if a "dead" team pulls back over the safety line, BJ apparently believes that they still do not have a meaningful chance of winning. In the OP's example, if a "dead" team pulls into the lead, they're still apparently supposed to be dead. This is pretty clearly absurd.

Incidentally, I was involved in a game in which a safe lead was overcome, but it was back in high school. We were down by 5, hit a three with 1.8 seconds left (so with two seconds left, we were still down by 5), stole the inbounds, hit a lay-up and got fouled, and won in regulation. I find it really, really hard to believe that no collegiate basketball team has done something similar.

What your post claims is that because the conditional probability is much higher, that the original probability is too low. But it's not. The conditional probability does not affect the original probability.

It does not remotely claim any such thing. Clearly your reading comprehension skills are lacking. My math skills are fine (I was a math major and went to graduate school in a quantitative field).

The OP is right, and this quote proves that James is confused. It's a basic fallacy of the statistically uninitiated; what he's missing is the idea of probabilistic "independence." If we're talking about a series of random events, what happens in the last 1:41 is independent of what happened between 3:00 and 1:41. It's exactly the same with rolls of the dice: you have the same chance, 1/6, of getting a 3 on the sixth roll regardless of what transpired on the first five; even if they were all 3's. Put another way, the ROUTE a team took to being behind 8 with 1:41 to go can't matter, but James thinks it does. If the team was always behind by about 8, or if the team had been down 17 1:20 earlier, can't matter: in either case the team has the same chance to win as of the 1:41 mark. If one of these teams is behind by less than a safe margin the other has to be also. It is surprising that Bill James would so misunderstand the concept of independent events.

it's not the same thing. whether the differences are significant is an open question.

think of it this way. you turn on the tv, your favourite channel, the nature channel, and you see the cheetah chasing the gazelle. good you say, perfect. now, is the gazelle safe? it's not just a function of the cheetah's speed and manoueverability, and the gazelle's speed and manoueverability, and the distance separating them. the history of the chase also matters. how much ground has the cheetah made up, how long has the cheetah been going full tilt. what is the gazelle's demonstrated maximum speed. what's about to happen is dependent on what has just happened.

same with the desperate b-ball end-of-game chase. it does matter who you've got out on the floor, how long they've been there, how hard they've been working, what the other team has in reserve, etc. it absolutely matters.

it's just a question of how much it matters.

i like to think that James had this in mind when he said the stuff he said, otherwise he is being a bit of a nitwit. the problem, though, he didn't explain any it. and it is important, because without at least touching on it he's leaving the question open to interpretation. [because one could say it matters the other way too, even if that cheetah is running out of gas, the fact that's he's closed from 50 ft to 5 ft may be infusing that gazelle with a sense of what's the point?]

I think a lot of you guys are missing the point.

Bill James pretty much makes his living coming up with reasons why weird statistical abnormalities can have devastating effects on a particular sports contest.

He isn't a mathematician, he is just a sports guy who wants to give us a different take on watching a ballgame.

I think that the main point of what James is saying here (and in his other publications), is that to truly enjoy the game, we need to know more than the TV talking heads (and slate columnists - I'm looking at you Robert Weintrob!) tell us about "toughness" and being "mentally tough" and "physical" and "agressive".

In the end, sport is the ultimate numbers game, and if you can't see the fun in getting lost in the numbers once in a while, maybe your not "mentally tough".

morphic guy,

The statistics don't "effect" a sports contest, they are a by-product of it. They may have explanatory power, and I guess this is what you wanted to say ... he likes to uncover neat new stuff that has susprising explanatory power. I hope my pointing this out doesn't have you feeling this is more of me missing the point.

And re the math quibble that some in this thread have, and I might have, depending, it's high school stuff, that's all. I don't think we need confer "mathematician" on a guy who's got Grade 12 Stats, and I don't think it's s stretch to assume James has got it. So if he made the mistake the top poster thinks he made, it's a really, really dumb one.

While I'm here I'll quibble with your "truly enjoy" thing too. I don't think James is wrapped up in himself and his work so much that he thinks we need it or something like it to truly enjoy the game. He's just fartin' around, that's all, and he's found a way to make a living at it.

Perhaps another way of making the point is that James' account is flat out contradictory. Suppose we pick some time t at which a lead satsifies the conditions for safety, then this lead is safe at t, and according to James, safe for all t'>t. Yet suppose we go to some such t', run the calculations, and the conditions are not met. James' account delivers the verdict that at t' the lead is not safe. So, at t', the lead is both safe and not safe. Reductio complete.

I know I'm kinda all over this thread so apologies if that's out of order or something, but ...

He could be implying that his formula works at time t if and only if there is no prior t" for which the formula produced (or would have produced) a safe result.

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Bill James: speak, man, say something! Throw us a frickin bone!!

I release my cheetah thing for free .. it's yours (for the book and all).

You say "So I turn on the TV with 5 minutes left and team X is ahead by 20, which is a safe lead according to your calculator. Team Y though makes a tremendous run and with 1 minute left the lead is one point."

This is your mistake. Of course Bill James would agree that the teams have a roughly equal chance of winning at that point. However, I believe that, according to James, your hypothesis here would never happen. James formula predicts not that the team that came back from 20 down in your scenario cannot win, but that your scenario will never happen.

I like to think of Bill's formula as if he's predicting predictable weather. For instance, in Kansas in July, it's always hot. But does that mean that if it was warm enough to be comfortable in a t-shirt outside today, you can assume the same for tomorrow? Most people would say yes (similar to James saying that coaches have a feeel for when to empty their benches). But imagine that Bill went through the historical record and charted the maximum temperature swings in Kansas in July to determine that as long as the temperature is at least 82 degrees today, you will be safe wearing a t-shirt tomorrow.

This is, in essence, his basketball formula. It conforms to the historical data, and will until a team comes back from a "safe" lead someday, at which time the formula will be recalibrated (or, more likely the game dismissed as a statistical anomaly)