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The paradox of the unexpected bouncer

A logical conundrum crops up in the cricket commentary

During the final Test match between India and England, the Indian fast bowler Umesh Yadav was giving the England batsmen a torrid time, bowling fast and short.

One of the radio commentators said: "Yadav always bowls one bouncer in an over, but you never know when it's coming," *

But can this statement be true?   

Suppose Yadav decides that he's going to bowl a bouncer on the final (sixth) ball of an over.  This means that after five balls there won't have been a bouncer, so - since we've been told there's always one bouncer per over - the batsman will KNOW that the bouncer must be coming in the sixth ball.  Which means it isn't unexpected.  In other words, the unexpected bouncer can't be on the sixth ball.

And since the bouncer can't be on the sixth ball, if the first four balls aren't a bouncer the batsman will know that the fifth ball will be the bouncer, which means that the fifth ball can't be an unexpected bouncer either.

This same logic can be used to show that there can't be an unexpected bouncer on the fourth, third, second or first ball of the over - in other words, the commentator's claim must be untrue.

And's the fourth ball of the over and Yadav runs in and... he bowls a bouncer.   And England's batsman Moeen Ali isn't expecting it.

This is cricket's equivalent of the paradox of the unexpected hanging, which has been baffling people for decades.


* If you aren't familiar with cricket terminology, a 'bouncer' is a short ball that goes above the batsman's shoulder height, and the regulation is that you can only bowl a maximum of one bouncer per over.   An over is a series of six balls bowled by a bowler.  Still baffled? There's a book called 'What is a Googly?' that explains it all in a bit more detail.