Powers of 9/10’s

An example of one of my favorite surprise calculations comes up in the context of quantum computing.  The idea is that if you have a process that gets things right 90% of the time and you repeat the process and the success or failure of one instance of the process has no effect on any other instance, the chance of being successful $n$ times is $(\frac{9}{10})^n$.  Thus this table:

Nine Tenths to the Nth Power

In just ten iterations, the chance of being correct ten times in a row is a mere 35%.  The chance of being correct 100 time in a row is  .003%.  This phenomena accounts for the following interesting graph.  Note that as $n$ increases, $x^n$ looks more and more like an “L”.  The vertical line is $x=.9$ and the dots locate powers of $.9$.

Nine Tenths to the Nth Power

One of the more interesting applications of this idea, which I think I read about in literature about random matrices, is about the random distribution of points in space. Suppose we distribute points randomly in the an n-space hypercube with unit sides.  The hypercube’s volume would be $1^n=1$ hypercubic units.  Now imagine a cube with sides $.9$ units inserted inside the unit hypercube.  It’s volume would be  $.9^n$ hypercubic units.  If $n=100$, a small dimension in this day and age of large data sets, $1^{100}-,9^{100}= .999973$ of the volume would be near the “surface” of the hypercube.  A random point(vector)  in the unit hypercube would have norm of nearly 1 (Use a n-space ball instead of a cube).  Interesting.