## May 2011

The probability of 20 consecutive bits being black is 2^{-20}, therefore the expected distance between two occurrences is 2^{20}.

There is a 0.5 probability of getting a black bit immediately after the 20 black bits, which should not be counted since it is the same bad bar.

Therefore the expected distance is 2^{21}.

Stephen Morris sent us a relevant link to this factor 2 "paradox": http://mathfactor.uark.edu/2009/07/morris-world-of-britain-2-proof-and-paradox (link resides outside of ibm.com).

Since we are seeking the distance between consecutive black bars, we should subtract the length of the bad bar, whose expected value is 21.

So the answer is 2^{21}-21 = 2,097,131.

This month we didn't ask an IBM-related bonus question, but we do have a bonus answer: IBM invented the bar code.

*If you have any problems you think we might enjoy, please send them in. All replies should be sent to:* ponder@il.ibm.com