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

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

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.

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About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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