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April 04, 2012

Line Calls Even Out in the Long Run, the Really, Really Long Run

You've heard tennis players and announcers say "The calls even out in the long run." But do they?

Review-system-challenges-at-Aussie-Open-3RSTBMB-x-largeThis is what's known as the Monte Carlo fallacy, or the Sucker's fallacy. It may be a good attitude to have as a player, but it's bad math.

If you assume that missed calls are random events (not a conspiracy or result of bias), then they are independent events. The classic example is a coin toss. If the coin is balanced, the odds of heads or tails coming up is 50% on each toss. 

Flip a coin 99 times resulting in 99 heads but no tails and the odds of getting a head or a tail on the 100th flip is the same as on all the rest: 50%. There is no cumulative counter somewhere in the ether directing a magic hand to even things out. 

TennistechYou're not "due". 

So, if you're in the club's 4.0 doubles final, or the Wimbledon finalist, and a questionable call goes against you -- tough luck, nothing has been added to your Karma bucket. Sorry. 

But, if you don't escalate to a verbal attack on the linespeople, you're a better person. That counts for something.

Top photo thumbnail from USA Today.

This post is part of the series, "A Skeptic's Guide to what 'Everyone Knows' About Tennis", click to read more. 

Reader Comments

Hi Jim,

Fun site.

Assuming line calls are random events, over the course of a player's career she should assume that the total set of calls will go approximately 50:50 in her favor. For example, let's have player A play 1000 matches over the course of her career: let's also assume that in each match there are 5 close calls, so she sees 5000 close calls during her career.

We can use the binomial distribution to calculate how likely it is that player A will get a certain number of calls in her favor, or calls going against her. With the aid of an Excel spreadsheet, I found that there's a 1% chance that A will get fewer than 2417 out of 5000 calls in her favor, or 48.34% of the calls in her career: and there's a 1% chance that she'll get more than 2582 calls in her favor, or 51.64%. Those are reasonably close to 50:50 in my book.

Of course, in an individual match, all the calls could go against A (chance of all 5 calls against A ~3%). Over the course of a season (500 calls), the 1%-99% likelihood range is about 45% to 55% of calls.

So you're never "due" on any call, in any match or in any tournament. But in the long run, in tennis, I think (assuming the umpires don't take a dislike to you) it does even out, pretty much. Unless you listen to John Maynard Keynes, who had a different take.

Thanks. Andrew.
I'm not remotely a stats guy . My least fav undergrad class.
I believe the stat guys would call your sample size too small.
Personally, I'd hit further inside the lines

From WikiP:

Gambler's fallacy arises out of a belief in the law of small numbers, or the erroneous belief that small samples must be representative of the larger population. According to the fallacy, "streaks" must eventually even out in order to be representative. [5] Amos Tversky and Daniel Kahneman first proposed that the gambler's fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic, which states that people evaluate the probability of a certain event by assessing how similar it is to events they have experienced before, and how similar the events surrounding those two processes are.[6][7] According to this view, "after observing a long run of red on the roulette wheel, for example, most people erroneously believe that black will result in a more representative sequence than the occurrence of an additional red",[8] so people expect that a short run of random outcomes should share properties of a longer run, specifically in that deviations from average should balance out. When people are asked to make up a random-looking sequence of coin tosses, they tend to make sequences where the proportion of heads to tails stays closer to 0.5 in any short segment than would be predicted by chance (insensitivity to sample size);[9] Kahneman and Tversky interpret this to mean that people believe short sequences of random events should be representative of longer ones.[10] The representativeness heuristic is also cited behind the related phenomenon of the clustering illusion, according to which people see streaks of random events as being non-random when such streaks are actually much more likely to occur in small samples than people expect.[11]
The gambler's fallacy can also be attributed to the mistaken belief that gambling (or even chance itself) is a fair process that can correct itself in the event of streaks, otherwise known as the just-world hypothesis. [12] Other researchers believe that individuals with an internal locus of control - that is, people who believe that the gambling outcomes are the result of their own skill - are more susceptible to the gambler's fallacy because they reject the idea that chance could overcome skill or talent. [13]

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