The Drunkard's Walk

Random events can be misleading.  Flip a coin.  It has a 50% chance of landing heads and 50% tails.  Now suppose it has landed heads 10 times in a row.  What are the chances it will land heads up on the next toss?  If it is truly an unbiased coin, the odds for a heads up will be 50%.  Past events have no effect on future events.  But what are the odds that a coin will land heads up ten times in a row? 

The chart below shows a plot generated from a random number generation program, Drunkard's Walk.  Try running the program Drunkard's Walk yourself and see what kind of trends you can generate from a random process. Enter any number in cell B8 (Count).  Press return.  Each plotted point is the sum of the previous plotted point plus the random assignment of a one or a minus one.  Are there trends?  Does it always look random?