# Math problem

January 27, 2013 |

Suppose a basketball team goes into a game making 37% of its field goal attempts. What are the odds that, during any 29-shot sequence, the team will go 0 for 29 in that sequence?

I’m coming up with 659,638 to 1.

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I get the same answer- actually 659,362.2 to 1. Using a spreadsheet. Here’s the detail for any auditors out there – after all, the Sainted ROnald Reagan said to trust bit verify!

try: odds each try: Cum odds: =One in:
1 0.63 0.63 1.6
2 0.63 0.3969 2.5
3 0.63 0.250047 4.0
4 0.63 0.15752961 6.3
5 0.63 0.099243654 10.1
6 0.63 0.062523502 16.0
7 0.63 0.039389806 25.4
8 0.63 0.024815578 40.3
9 0.63 0.015633814 64.0
10 0.63 0.009849303 101.5
11 0.63 0.006205061 161.2
12 0.63 0.003909188 255.8
13 0.63 0.002462789 406.0
14 0.63 0.001551557 644.5
15 0.63 0.000977481 1,023.0
16 0.63 0.000615813 1,623.9
17 0.63 0.000387962 2,577.6
18 0.63 0.000244416 4,091.4
19 0.63 0.000153982 6,494.3
20 0.63 9.70088E-05 10,308.3
21 0.63 6.11155E-05 16,362.5
22 0.63 3.85028E-05 25,972.2
23 0.63 2.42568E-05 41,225.6
24 0.63 1.52818E-05 65,437.5
25 0.63 9.6275E-06 103,869.1
26 0.63 6.06533E-06 164,871.5
27 0.63 3.82116E-06 261,700.9
28 0.63 2.40733E-06 415,398.2
29 0.63 1.51662E-06 659,362.2

• greylocks says:

If we were talking dice or coins and there were exactly 29 trials (shots) and no more, then the probability of missing all 29 is .63^29 which is about .00000152 which comes out to 1/659,362, according to Windows calculator.

However, if they took more than 29 shots, then the probability of missing some 29-shot sequence is somewhat higher. The calculations are complicated and frankly not worth the effort, but there’s probably a free online applet somewhere that will do them for you if you’re all that interested.

You can further complicate matters by calculating the probability of missing at least 29 in a row, as opposed to exactly.

And as others have pointed out, there’s a laundry list of possible reasons why the historical average may not have been meaningful for this game. There are also possible reasons why the historical average may be bullshit (e.g., a shooter who got super-hot for a few games before returning to form.)

2. John Protevi says:

Yes, but you’re assuming — and it may very well be a good assumption, but it would need to be empirically verified — that the defensive capabilities of the team against which they went 0 for 29 are roughly the same as those of the teams against which they shot 37%.

• Bill Murray says:

and ignore that different players have different chances of making shots, and probably more importantly different shots have widely varying chances of success. Sure using the mean minimizes error if you know nothing else, but it’s usually not that hard to get more information, particularly data on the original context of of the offensive and defensive percentages as John P. says

• mpowell says:

I think if you looked at it carefully, you’d find that the variation in the statistics of making different kinds of shots leads to a higher likelihood of streaks. And you can combine that with the fact that you will certainly see correlation (both negative and positive) for stretches of time as a team’s defense figures out how to shut down an opposing offense for a particular set of things they are trying to do (and then the offense figures out how to beat that defense, leading to negative correlations in other cases).

In other words, basketball is not a random walk process.

• Bijan Parsia says:

In random processes, streaks are not uncommon (depending on the size of the domain). Oh! And there’s a paper!

• Colin Day says:

Yes, the calculation only holds if one assumes that the shots are probabilistically independent, which might not hold for human shooters.

• Bijan Parsia says:

The paper I references makes a reasonable case that they can be so regarded at least enough to cast strong doubt on any “streak” phenomenon.

• Colin Day says:

But it raises questions as to anti-streakiness. The article (Table 2) gives a z score of -1.88 for Andrew Toney’s streakiness. The p value for this is 0.06, which is greater than the alpha of 0.05. But we’re not testing methods of brain surgery, so why should we have such a low alpha?

• Bijan Parsia says:

Sorry, I went back and looked and I’m afraid I’m not following your concern. Spell it out for me?

• Colin Day says:

On table 2 of the article you cited, there is a z score of -1.88 for Andrew Toney. The probability that one would have such a score given that his shots are independent is 0.06 (p=0.06). For medical testing, taking alpha=0.05 or lower is prudent, but should we err so much on the side of caution here? Also, since the score is negative (Toney has more runs than expected), Toney is anti-streaky, i. e. more likely to miss a shot if he made the previous shot.

• Bijan Parsia says:

Significance lines are “merely” conventional, so, yes, it’s clear that there’s some correlation there. And yes, he’s antistreaky.

But so? Both anti-streaky and streaky looking patterns are possible in a random process.

Isn’t the interesting thing about Toney (cf pg 303) is that he was perceived as a streak shooter when his record is moderately antistreaky? That should at least give us some pause, right?

3. Mudge says:

You need to factor in the defensive prowess of Eastern Michigan. And the 3 point versus 2 point percentages. But it’s still a big number.

4. charles pierce says:

Odds, hell. How to you take the 23rd shot?

5. Ben says:

I think this reduces the shot to something isolated like a coin flip. I see they ended up shooting 13.x% for the game, which is definitely far lower than their season average, but picking a given 29 shot sequence strikes me as cherry picking the exceptional(ly bad).

More simply, if you are flipping a coin with a 37% chance of getting heads, and you get tails 29 times in a row, i trust your calculations….so on a thought exercise as to that type of thinking/calculation…good on ya. But in basketball, the FG% – both offensive (and defensive for the opposing team) – are aggregated stats…so a given run is just that…a sequence that contributes to the larger FG% calcs (that are aggregated/ongoing).

Yes it was exceptionally bad, but i’m not sure it can be reduced to such a specific odds calculation. Someone more intelligent in the world of complex stats can probably correct me on this.

(note: you see this all the time in sports commentary…”at this rate, they’ll rush for 800 yards” (after a big first quarter) etc etc…. not surprisingly, by the end of the games, some of these exceptional windows come much closer to expected deviations around their seasonal averages.

6. Yes, 659,362 to 1 is the odds of missing that particular 29-shot sequence. So if you had asked before the game, “What are the odds that Northern Illinois would miss shots 2 through 30 against Eastern Michigan,” that’s your best estimate. If you want to know what’s the chance that they’ll miss 29 straight shots at any point in the season, you have to consider that there are, what, a couple thousand shots in one season, each of which starts a 29-shot sequence. So the odds that Northern Illinois misses 29 straight at some point in the season are more like 329 to 1. Some of their opponents are better defensive teams than others, which makes 29 straight misses even more likely. Multiply by the number of teams with lousy shooting percentages each year, multiply by the number of years people have been keeping track of college basketball statistics, and it was bound to happen sooner or later.
But yeah, NIU alums have an embarrassingly bad basketball team to cheer for.

• GFW says:

+1 for explaining how things that are statistically unlikely at any given point are ultimately close to inevitable.

• Njorl says:

“Everything not forbidden is compulsory.”

-Murray Gell-Mann, a la TH White

7. Vance Maverick says:

The math is a bit more complicated. See this Stack Overflow discussion for a start.

• LosGatosCA says:

That’s impressive. My head started hurting before the first paren.

Solve this one: What are the odds of an NBA point guard continuing to feed the ball to a shooter who goes 0-17 in a game AND the point guard and the shooter are the same person? And the team actually wins the game?

What does de Moivre have to say about that?

• efgoldman says:

My head started hurting before the first paren.

Mine started hurting at “Suppose….”

8. kgus says:

Actually, the odds are much worse than that. The proof is obvious.

• GFW says:

But this comment box is too small to contain it?

• Doug says:

And so it remains unsolved for the next couple of centuries. I think I’ll stop there, for today.

• Njorl says:

No.

There are multiple ways to interpret the problem, but those interpretations yield higher probabilities, not lower ones. Your “obvious” proof is obviously wrong.

• efgoldman says:

The proof is obvious.

Intuitively obvious to the most casual observer?

• Bill Murray says:

shouldn’t you have used the name Fermat when saying something like that

9. Green Caboose says:

The 1995-6 Chicago Bulls – they of the 72-10 fame – had a team 3 pointer FG percentage of just over 40%.

Yet, in game 5 of the finals they threw 20 consecutive bricks from behind the 3 point line.

Somehow I think that was more improbable than a weak college team missing 29 in a row.

10. Dano says:

This is similar to my issue with that paper that came out that said ‘streaky shooters? Not so much!’.

If you play basketball, you know you want to get it to the person with the hot hand. That’s how we played.

Making them is psychological, just like missing. I saw Mark Aguirre make 9 in a row one game, and after 6 or 7 he thought he couldn’t miss. And for a couple more, he couldn’t. Then he took a J off balance and missed, then he bricked a couple, then coach made him sit to get his head in the game.

Best,

D

• Bijan Parsia says:

You mean this one??

If you play basketball, you know you want to get it to the person with the hot hand.

Isn’t the question whether you are right to so want?

Making them is psychological, just like missing.

What’s the evidence for that?

I saw Mark Aguirre make 9 in a row one game, and after 6 or 7 he thought he couldn’t miss. And for a couple more, he couldn’t.

How do you know this? What makes the 6 or 7 different from the subsequent 2 that were followed by:

Then he took a J off balance and missed, then he bricked a couple, then coach made him sit to get his head in the game.

Yeah, this isn’t so very convincing. The psychological state might have no causal effect whatsoever. Indeed, it could be that treating him specially would lower his overall performance (it’s easy to imagine many scenarios e.g., he’s in fact better than everyone else so pulling him for a negative “streak” is counterproductive).