Thursday, December 30, 2010

Analytics and Cricket - IV: The Great Indian Coin Toss

We'll end this year with the humblest of analytical models - the coin toss. It is an important benchmark. After all, if your predictive business model can consistently outperform a coin-toss approach, then that could be a big deal in many practical situations. So what do we make of the Indian cricket captain Mahendra Singh Dhoni's (MS for short) performance with the coin? He's lost 13 of the last 14 trials!

A coin toss can be a big deal in cricket, since a 'win' allows you to decide whether to bat or bowl first. A 'flat' wicket means it's a great one to bat on and make best use of it, and the opposition gets to play on the same pitch after potential wear and tear. A 'sticky wicket' or overcast conditions on a 'green' pitch means bowling first could be a great option since batting will be difficult for the first few hours due to the 'swing' and 'seam' movement potentially available to the bowlers.

Die-hard cricket fans like me and players are among the most superstitious in the world due to the long and complex nature of the game. MS gets blamed for "losing" the toss and he's even asked for tips on improving his record :) Useful analytical models are nice to have, but they could go horribly wrong, especially when applied to cricket ... Before the sports fan begins to question his faith in science and even doubt the fundamental idea of Bernoulli trials and the law of large numbers, we note that if MS had lost 14 tosses in a row, that would have been an extreme "achievement" since the probability of that happening would have been roughly 60 in a million, and that did not happen. Phew! that counts as favorable evidence.

With the India-South Africa cricket series tied at 1-1, and with one test match to go, we have no choice but to seek solace in the scientific estimate that our fearless captain still has a 50% chance of winning the toss in Cape Town. I know that doesn't sound encouraging. But there's got to be a point in time when nature is going to bring that win-loss average back close to 50%. Will that happen in 2011? who knows ...

In 2010, MS had several ways of losing 13 of the 14 tosses. More simply, he had 14 ways of winning exactly one toss. We know that the probability of winning m of n tosses follows the Binomial distribution, and we can find out online here that the chance of losing 13 out of 14 is still tiny, at 0.00085. In other words, the chance of him winning 2 or more tosses in 2010 was greater than 999/1000, and yet that did not happen!

Like most great teams, this current Indian cricket team does not depend much on the outcome of the toss. Put into bat on a green, bouncy wicket under overcast conditions, they still managed to defeat RSA in 4 days and displayed amazing skill and resilience in the process. Still, it wouldn't hurt to begin the final match between No.1 in the world (India) and No.2 in the world (RSA), starting on Jan 2, 2011, by winning the coin-toss. If MS loses that toss, then the probability of this extended streak over 15 trials would be around 0.0004, i.e., 50% less than the already dismal number he is at today. Surely, that's unlikely, right? Let's see. What is the probability of the sequence that ends with him winning the toss on Jan 2, i.e., the chance that he wins exactly 1 of the first 14, and then win the 15th? Sadly, that's not very different. Delving into the past does not help the Indian sports fan, and talking to statisticians would not help since none of them wants to see such a rare streak end :)

It is better to look forward to the new year, where 2010 is done and dusted. We can say it again: MS has a 50% of winning the next toss, and relatively speaking, that looks so much more promising and simpler to comprehend.

Happy New Year and Go India!


  1. I'm not a cricket fan, but my guess is that a fair number of cricket matches are played each year world-wide, and a fair number of captains call quite a few consecutive coin tosses before being replaced. So the "population" of sequences of 14 consecutive calls by the same captain is probably fairly large, which means that the odds of any individual captain having the bad luck of MS are low, but the odds of some captain somewhere being that unlucky are probably fairly high ... which I suppose is no consolation to fans of India's team.

  2. Good point you pointed out. Yes, we love to shine the spotlight on a particular instance of occurrences and speculate on its failure to adhere to the rules :) We all believe in the laws of probability, but real events can test that faith.

    And the last observation was spot on!

    As an extreme example: Suppose we deliver a stochastic decision model to a customer whose success is largely tied to the law of large numbers. If we had many customers, or if it was used to run over many data sets and the averaged results are what count toward a success, no problem. but when we do the first 'pilot' for one customer on a small number of 'test' data sets, we better have some plan B's, i suppose.


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