Math and Logic Errors

[OldPhiKap]

I've never played, but aren't their pocket aces likely (at least 50% chance) to be cracked? If most players stay in, then doesn't the person with pocket aces have less than a 50% chance of winning?

I'm not sure anyone would be stunned when a 45% FT shooter misses. Hack-a-Shaq may be prudent.

Correct, the more people who stay in, the odds go down. My figure was against a single player. I think it drops down to around 52% (0.85^4) if you get four callers instead of one. If you are playing ten-handed and get nine callers, I think it drops down to about 23% (0.85^9) to still win.

Although perhaps I have just displayed a mathematical fallacy in my thinking.

[Jeffrey]

The main thing I would want to ask Century Man is, what part of John Cage's Organ2/ASLSP was his favorite?

The first thing I would ask Century Man is, "who's your daddy?".

[Jeffrey]

Correct, the more people who stay in, the odds go down. My figure was against a single player. I think it drops down to around 52% (0.85^4) if you get four callers instead of one. If you are playing ten-handed and get nine callers, I think it drops down to about 23% (0.85^9) to still win.

Although perhaps I have just displayed a mathematical fallacy in my thinking.

Thanks, it sounds like a lot of fun!

[OldPhiKap]

The first thing I would ask Century Man is, "who's your daddy?".

Battier, obviously.

[rsvman]

OK, so on a related note, I want to know if this math/logic is correct or not.

Instead of JJ, let's talk about George Hill. When he went to the free throw line with 4.7 seconds left in game 1 of the Finals, the announcers said he was an 80% FT shooter. For the purposes of this discussion, let's assume that number is correct.

Is it or is it not true that the a priori chance that an 80% free throw shooter will make both free throws is the product of the individual odds? In other words, 0.8 X 0.8, or only 64%? If it IS true, then aren't the chances that JJ will make two in a row 0.9 X 0.9, or only 81%? If true, that would mean, essentially, a 1 in 5 chance that he doesn't make both.

And if so, I would argue, "not rare."


If this logic/math is incorrect, please explain how/why, so I can be enlightened. Thanks.

[Jeffrey]

And if so, I would argue, "not rare."

IMO, it all comes down to whether we are discussing a particular JJ FT or an entire game of JJ's FT's. IMO, if we are discussing a particular FT, then it is unlikely (only 10% chance) JJ misses. If we are discussing an entire game, in which JJ shoots 10 FT's, then it's probable JJ will miss one. Play-by-play announcers are usually discussing the current event.

Clearly, this is debatable and being critical of announcers for calling a JJ FT miss "rare" is probably not appropriate. Aren't there many better reasons to criticize announcers?

[HereBeforeCoachK]

OK, so on a related note, I want to know if this math/logic is correct or not.

Instead of JJ, let's talk about George Hill. When he went to the free throw line with 4.7 seconds left in game 1 of the Finals, the announcers said he was an 80% FT shooter. For the purposes of this discussion, let's assume that number is correct.

Is it or is it not true that the a priori chance that an 80% free throw shooter will make both free throws is the product of the individual odds? In other words, 0.8 X 0.8, or only 64%? If it IS true, then aren't the chances that JJ will make two in a row 0.9 X 0.9, or only 81%? If true, that would mean, essentially, a 1 in 5 chance that he doesn't make both.

And if so, I would argue, "not rare."


If this logic/math is incorrect, please explain how/why, so I can be enlightened. Thanks.

You are correct sir.

[Mal]

Perhaps rarity can be relative, such that it is rare for JJ to miss a free throw compared to typical players (cases).

Ding ding ding! I can't believe it took all you intelligent people four pages of comments to get to this point. This isn't really about numbers, or at least not only about numbers. It's about words.

Of course it's relative. You're talking about communication from television announcers during the course of a basketball game with at least ten participants, not pure statistics. Everything has to be viewed through that lens. If an announcer says in a fairly flat tone "A rare miss from the line by Redick" there's no underlying judgment as to some statistical definition of what "rare" means in a vacuum. What would we have them say instead? "A statistically improbable event right there, in the case of a single free throw, although over the course of a full 48 minute game, a likely occurrence at some point." No thanks, Professor. There's limited time, and a limited vocabulary. How many words or phrases out there mean "sort of rare but not really" if that's what 10% for an individual occurrence means? "Medium rare?" "Kind of rare?" "Uncommon?"

I agree that if the announcer acts totally shocked and exclaims emotionally something like "I can't believe he missed! That's truly a surprise. A rare miss by Redick" it seems a bit innumerate. But I think that's...umm...announcers don't act like that all that often.

[freshmanjs]

I agree that if the announcer acts totally shocked and exclaims emotionally something like "I can't believe he missed! That's truly a surprise. A rare miss by Redick" it seems a bit innumerate. But I think that's...umm...announcers don't act like that all that often.

It plays out in different ways. I just watched a video of Michael Westbrook missing a free throw in the 4th quarter of a game in which he took 20+ free throws. He missed 2 in the 4th quarter. Announcer said "It's been a real struggle for Westbrook in this quarter, missing 2 free throws." When, the chances of him missing 2 free throws in a quarter in which he attempted 5 is about 1/4. So, in a game, you should not be at all surprised that it happened in a quarter. It's not "a real struggle," it's just an outcome based on probability, and not a surprising one.

[Reilly]

OK, so on a related note, I want to know if this math/logic is correct or not.

Instead of JJ, let's talk about George Hill. When he went to the free throw line with 4.7 seconds left in game 1 of the Finals, the announcers said he was an 80% FT shooter. For the purposes of this discussion, let's assume that number is correct.

Is it or is it not true that the a priori chance that an 80% free throw shooter will make both free throws is the product of the individual odds? In other words, 0.8 X 0.8, or only 64%? If it IS true, then aren't the chances that JJ will make two in a row 0.9 X 0.9, or only 81%? If true, that would mean, essentially, a 1 in 5 chance that he doesn't make both.

And if so, I would argue, "not rare."


If this logic/math is incorrect, please explain how/why, so I can be enlightened. Thanks.

I guess the odds for the 90% shooter -- before the first shot -- are 81% he makes both, an 18% chance he makes 1-of-2, and a 1% chance he misses both. Announcers should not use words ("rare"/"surpise") and just stick to the numbers. So when the shooter misses both, the announcer says "we just saw something happen that we only expect to happen 1 out of every 100 times that Shooter steps to the line for two FTs."

[HereBeforeCoachK]

I guess the odds for the 90% shooter -- before the first shot -- are 81% he makes both, an 18% chance he makes 1-of-2, and a 1% chance he misses both. Announcers should not use words ("rare"/"surpise") and just stick to the numbers. So when the shooter misses both, the announcer says "we just saw something happen that we only expect to happen 1 out of every 100 times that Shooter steps to the line for two FTs."

Considering that many sports announcers have a hard time figuring what 2 out of 4 is, I doubt we'll see that kind of insight any time soon.

[BLPOG]

...

Is it or is it not true that the a priori chance that an 80% free throw shooter will make both free throws is the product of the individual odds? In other words, 0.8 X 0.8, or only 64%? If it IS true, then aren't the chances that JJ will make two in a row 0.9 X 0.9, or only 81%? If true, that would mean, essentially, a 1 in 5 chance that he doesn't make both.

...

I'm not really sure what you mean by "a priori" in this case since a person's free throw percentage is an empirical calculation.

That said, what you basically asking is whether consecutive free throws are independent events. If they are, 0.8 * 0.8 is correct for calculating the odds of making both. If, for guaranteed-two-shot-attempts sets of free throws, the percentage was not 0.64, that would indicate that they are not independent events. Similarly if the percentage of made shots for the first and second shot differ.

I suspect that the percentage does differ, but only by a small amount, and more in the final minute of a game than at any other point. If it differs considerably throughout games, that would be evidence of a "hot hand" effect. If it only differs at the very end of the game, it could indicate something like intentionally missed free throws.

[camion]

I'm not really sure what you mean by "a priori" in this case since a person's free throw percentage is an empirical calculation.

That said, what you basically asking is whether consecutive free throws are independent events. If they are, 0.8 * 0.8 is correct for calculating the odds of making both. If, for guaranteed-two-shot-attempts sets of free throws, the percentage was not 0.64, that would indicate that they are not independent events. Similarly if the percentage of made shots for the first and second shot differ.

I suspect that the percentage does differ, but only by a small amount, and more in the final minute of a game than at any other point. If it differs considerably throughout games, that would be evidence of a "hot hand" effect. If it only differs at the very end of the game, it could indicate something like intentionally missed free throws.

I can state without fear of contradiction* that the first and second free throws are not independent events. Being too lazy to run the numbers I would expect second free throws of to be made at a higher rate than the first free throws since the first free throw can be practice for the second.


* It's not that I don't expect contradiction; I just don't fear it.

[freshmanjs]

I can state without fear of contradiction* that the first and second free throws are not independent events. Being too lazy to run the numbers I would expect second free throws of to be made at a higher rate than the first free throws since the first free throw can be practice for the second.


* It's not that I don't expect contradiction; I just don't fear it.

Here is a ton of data on this...it does indicate that FT 2/2 is more likely to be made than 1/2 or 1/1. Interestingly, (maybe?) attempt 2/3 is made with a higher % than either 1/3 or 3/3. Another thing I found interesting: JJ Redick shoots 1/2 better than 2/2.

https://fansided.com/stats/player-shooting-breakdowns/

[jimsumner]

It plays out in different ways. I just watched a video of Michael Westbrook missing a free throw in the 4th quarter of a game in which he took 20+ free throws. He missed 2 in the 4th quarter. Announcer said "It's been a real struggle for Westbrook in this quarter, missing 2 free throws." When, the chances of him missing 2 free throws in a quarter in which he attempted 5 is about 1/4. So, in a game, you should not be at all surprised that it happened in a quarter. It's not "a real struggle," it's just an outcome based on probability, and not a surprising one.

What are the odds that it would take 80 minutes for someone to suggest that perhaps, perchance, you were referring to a different Westbrook than Michael?

[freshmanjs]

What are the odds that it would take 80 minutes for someone to suggest that perhaps, perchance, you were referring to a different Westbrook than Michael?

Oops, you're obviously right!

[Reilly]

... I would expect second free throws of to be made at a higher rate than the first free throws ...

... it does indicate that FT 2/2 is more likely to be made than 1/2 or 1/1 ...

Al McGuire when announcing games used to often bring up if the player *makes* the first free throw he's much more likely to make the second (as compared to his usual percentage, I guess, though Al didn't explain his theory in detail).

[rsvman]

Thanks, guys. I suspected that my math was good for independent events, but I also suspect, as many of you do, that free throws taken back-to-back are probably not strictly independent. Still, I think it provides a fairly reasonable expectation of the odds, which to the casual observer, likely seem surprisingly low.

In other words, I'm willing to bet that both the announcers and likely a large proportion of the viewing audience expect more double makes than actually occur. When the casual viewer who is not thinking about math hears that, for example, a player is an 80% free-throw shooter, they fully expect the guy to make both free throws. They may even, if not overtly, expect there is an 80% chance that the player will make both free throws, when the actual probability is much closer to 65% than it is to 80%. Thus, they are likely to be surprised when they needn't be, and they are likely to be disappointed fairly frequently.

All this is just conjecture on my part, though.

[HereBeforeCoachK]

Al McGuire when announcing games used to often bring up if the player *makes* the first free throw he's much more likely to make the second (as compared to his usual percentage, I guess, though Al didn't explain his theory in detail).

Yes, and of course that injects emotion and subjectivity into what we've been discussing as an objective math equation. I think McGuire's theory is probably more correct on players who shoot few free throws, and not so much on guys used to shooting foul shots, who handle the ball a lot, etc.

That subjectivity thing is a factor. I mean, Laettner was probably what, a 75-80% shooter? What was he in the clutch? 99%?

[jimsumner]

Al McGuire when announcing games used to often bring up if the player *makes* the first free throw he's much more likely to make the second (as compared to his usual percentage, I guess, though Al didn't explain his theory in detail).

I liked Al. But I always got the impression he was more of an intuition kind of guy than an analytics kind of guy.