This is mostly a post about cricket but also an occasion for desultory thoughts on confirmation bias, selection on outcomes and causality. India is reaching the end of a long tour of South Africa and India captain M.S. Dhoni proceeded once again to lose the toss in the first test, arguably costing India the match and thus a shot at winning the series outright.
Dhoni’s dreadful run of luck with tosses has inspired a whole sub-genre of sporting literature with themes ranging from – “Why Can’t Dhoni Win” , to “Do The Laws of Cricket Require the Captain to Toss?” to the more creative “Is Racism Behind Dhoni’s Losses” and “Dhoni Loses Tosses Because He Does Not Practice Them”.
Naturally A Spoonful of Win can hardly be expected to keep silent on this issue of national importance (well at least to one nation).
I began, as do all well trained PhDs, with a quick Google search hoping to find a ready made solution. The question I wanted to answer was whether Dhoni was actually losing more tosses than he should or whether we only remembered the losses? Confirmation bias normally explains these things and there seemed no reason to doubt otherwise here. Unfortunately step one drew a blank because surprisingly no one seemed to have done the statistical legwork for me (that or my Google skills are fading fast).
Step Two was a quick visit to Cricinfo – the source of all wisdom on matters cricketing. A few minutes figuring out how the oracular StatsGuru tool on Cricinfo works and soon I had the information required. As the two tables suggest, confirmation bias certainly seems to be at play as far as ODI tosses go. Dhoni wins about 47 percent of his tosses. Test matches are a different story – he wins about a third of all tosses. A coin flip record as bad or worse than that is likely to happen only about 7.5 percent of the time in a sequence of 24 trials. Improbable, but even so evidently Dhoni wouldn’t fail a standard hypothesis test for normalcy (where normal here means binomial).
But lets dig a little deeper. First, how unexpected is a 33 percent win loss record? Sure it will happen with a small probability for a randomly selected captain who as tossed at least 24 times, but Dhoni is not randomly selected. On the contrary we’re only talking about him because he loses a lot of tosses. With at least 6 cricket captains out there who hang around for a reasonable amount of time (Pakistan for instance is automatically excluded), you’re much more likely to find one who calls wrong often.
Is that the end of the Dhoni toss mystery then? Well maybe, maybe not. It turns out that MS has also quite recently lost 9 tosses in a row. In fact it was this spectacular run of bad luck that’s probably given him his reputation for coin flip uselessness. So what are the odds of that happening? Well first lets refine the question. The odds of 9 losses out of 9 tosses are very small but once you’ve tossed 24 times, there are all of 16 sequences of length 9 in there. That pushes up the chance of seeing a losing run (just as a hot streak in basketball is more likely to happen than most people think). A bit of dubious, half forgotten high school math later I calculate that the chance of a 9 streak in 24 tosses is about 3 percent. So is that the answer we want?
Not really. What we really want to know is – Given a man is unlucky enough to lose 16 tosses out of 24, what are the odds that a 9 game losing streak will also be in there somewhere? This is of course much more likely than an unconditional losing streak and a little Bayes Ruling later (possibly the first time Bayes has answered a question I have actually been interested in knowing the answer to), it turns out the probability of 9 losses in a row, given 16 losses out of 24 is a lot higher: just under 14 percent. Not so surprising then.
So there it is. Dhoni is unlucky but not as much as we think (Confirmation Bias and Selection on Outcomes). So whats the causality story? Well that has nothing to do with cricket but a lot to do with inference. If you look hard enough at a population you will find a few outliers. And then if you look a little more you may find differences between some characteristics of the outliers and the everyone else. All too often we then see the argument Random characteristic X strongly correlated with outcome Y, watch out for Y. (Normally sitting in a story in the New York times headlined something like “Medical Study Finds Listening to Brahms at Two in the Morning Improves Your Sex Life”).
Thus after a great deal of statistical analysis on a panel of data composing 3 captains and many tests I conclude that if you’re a wicket-keeper captain coins just don’t like you.
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