## Friday, 30 August 2013

### Two simple maths / cricket problems inspired by Aaron Finch

Aside: To comment on my recent Gresham College lectures please go to this blog entry. My most recent Gresham College lecture was on Monday 15 April.  I talked about proof, by human and by computer.  The lectures can be viewed on the Gresham College website.

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These problems were inspired by Aaron Finch's great innings last night.  Apologies that some knowledge of cricket is required.

In a Twenty20 match Australia bat first and score 261.  England's openers are A and B, who are both remarkably consistent.  A plays every ball he faces for 3, while B scores 2 off every ball he receives.  In an equally unrealistic manner, the Australian bowlers never bowl no-balls.

So the openers score an average of 2.5 runs per ball, and therefore over 20 overs England will expect to score 300.  But after how many balls will England win the match?

Second problem: Unfortunately England's top batsman, A, is unavailable for the next match and is replaced by C, who scores a single run off every ball he faces.  Clearly a loss of two runs per ball will make a big difference to England's total score.  How many fewer runs will they make off 20 overs when C, rather than A, opens the batting with B?

(Photo by Supun47 from Wikimedia Commons)

These questions were motivated by the observation that Finch faced more than his fair share of the bowling last night.  So was luck a factor in his making such a high score?  No, it wasn't, because a batsman who is scoring lots of boundaries will have much more of the strike than a batsman scoring in singles!

1. If A faces first and scores 3 off that ball, he won't face until ball 1 of the next over. Therefore the average score per over will be 3+2*5=13, so Australia will win by 1 run!
If B faces first, then the first over will go for 12, and A will then face ball 1 of the second over, and the above applies - a total of 259, and Australia win by 2 runs!
The scenario where both alternate will never happen, unless they both score odd numbers of runs.

If C replaces A, then if C faces first, score is 220, and if C faces second, score is 221, as two fewer runs will be scored in overs where C faces compared to A facing.

2. very interesting site when i play your game then i think more deeply about your game..... and i just realize my brain will be sharped.....

3. Nice to see your blog it's really interesting topic. i am really excited about that game. and good answer there by first one.