Another neat way to look at it--the probability of any given element is the probability of it being swapped in times the probability it isn't swapped out in each successive item. For A[p]:

(1/p)((p)/(p+1))((p+1)/(p+2)) ... ((n-1)/n))

e.g. the fifth element's probability in an eight-element list is

1/5 * 5/6 * 6/7 *7/8

Since the numerator of each successive term cancels out the denominator of the last, the product is always 1/n.