Thursday, March 27, 2008

Sequences in cricket scores

A few days ago, I wrote to David Barry of Pappus' Plane with a nerdy question:
Are there any recorded sequences of scores by batsmen that form mathematically interesting sequences? Like for example a Fibonacci sequence (a batsman recording 0 1 1 2 3 5 8 13 21 34 in a test series?) Of course, this would be pure coincidence, but still, it would be very satisfying.
David wrote back:
The sequences you're talking about are very rare. The longest Fibonacci type sequence (ignoring zeros) has length 4, and has been achieved by 23 players (possibly more if there were any in the latest Eng-NZ Test). The earliest of these was Billy Midwinter, who went 4, 4, 8, 12 in the middle of the 1881/2 series. I'm not sure if there are any 1,1,2,3's.

Arithmetic progressions: Cameron Cuffy went 0*, 1, 2, 3*, 4 early in his career, spread over a few series.

Geometric progressions: Nine players have had a GP of length four, the first being by Jack Worrall with 1, 2, 4, 8 across a couple of series in 1888.

There are almost certainly longer sequences in first-class cricket, but I don't have all first-class scorecards in my database yet (I've got about 14000 still to go I think).

That the longest such sequences should be so short is not too surprising, given how many possible scores a batsman can make in an innings. Across all batsmen, a score of zero has probability about 0.12, and a one has probability around 0.05. And 0.05^4 is 1/160000.
So I wrote back:
Thanks very much! So pretty much as I expected. I'd have been very surprised if something really extended or interesting had shown up (it'd be cool to get a sequence of prime numbers for instance). I'm guessing extended sequences of odd numbers and even numbers must be much more frequent.
And David responded:
I have a correction to my last email: only six players have had a geometric progression of length 4. Some others:

Mark Taylor ended his career with a sequence of eight prime numbers (61, 3, 59, 29, 7, 19, 2, 2), a record since equalled by Mathew Sinclair.

Adam Gilchrist has the record for the longest streak of even scores at 19: 42, 2, 2, 6, 44, 2, 0, 86, 12, 2, 24, 12, 0, 144, 12, 0, 64, 0, 102*.

Waqar Hasan has the record for the longest streak of odd scores at 14: 5, 23, 81, 65, 49, 29, 97, 9, 53, 7, 7, 11, 7, 9.

As you'd expect, the record even streak is longer than the record odd streak, because ducks are so common. [note from SC: You all know zero is an even number, right?]
Fascinating stuff, don'tcha think?

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