Let p0,p1,… be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let Ai denote the event that the number i has been selected and that it is in the same place in both lines. Prove that the events A_i \;(i\equal{}1,2,\ldots) are mutually independent, and P(A_i)\equal{}p_i.
T. F. Mori probabilityprobability and stats