8.  The DERIVE command comb(n, r) simplifies to give you the number in position r in the nth row of Pascal's Triangle and use them to determine the average distance moved in a 16-step random walk.  According to the relationship we obtained for the average length of a random walk of given number of steps, how far would we expect to go, on the average, in a 16- step random walk?

Use the program DERIVE and put in the equation comb(n, r)  to find the number in the place of the 16 step random walk, replace n with 16 and r with the number in that row of Pascal's Triangle; such as 0, 1, 2, 3, 4,.....

The numbers in the walk are

1, 16, 120, 560, 1820, 4368, 8008, 11440, 12820, 11440, 8008, 4368, 1820, 560, 120, 16, 1

To find the number of walks from the origin is to find if n is even or odd.  If n is even, then the number of steps will be even.  If n is odd, then the number of steps will be odd.  By finding the even and odd you can use this to find the number of steps between by starting at 0 and going up to the highest point of 16.  

By doing this the even is 2, 4, 6, 8, 10, 12, 14, 16;  and odd is 1, 3, 5, 7, 9, 11 ,13 ,15

Take these numbers and double each and add the answer times the number of walks away from the origin.  and divide by 2^16=65536.  to find  the average distance traveled in the random walk

The equation will look like this (2*16+32*14+240*12+1120*10+3640*8+8736*6+16016*4+22880*2+12820*0)/65536

205920/65536=3.142089844

But the most traveled area on the walk would be at 0.  This is because the probability of flipping a coin is to get one tail for every head and this would cause most walks to be at 0.

By. Mitch Owens