My office is doing a fantasy football draft.  Everyone in the league
is going to pull one of sixteen numbered ping-pong balls out of a bag.
 Another person and I randomly selected the order in which people will
draw their ball from the bag, using these same ping-pong balls. 
However, there are people in this group who believe there is an
advantage to drawing a ball first or early, rather than late or last. 
Does everyone have the same odds of getting the #1
ball, or is there an advantage to drawing first?

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Clarification of Question by justaguy1969-ga on 29 Aug 2006 12:13 PDT

We are determining the order in which we will draft.

Might I suggest that each draw cancels the other out? 1:16, 1:15, etc.
Regardless, the numbered ball is allocated to the team, not to the
person drawing the ball. Others learned in combinations and
permutations may beg to differ and provide a better answer.

e.g. Take the game show example. The host asks the contestant to
choose one of three doors from which the prize is concealed. The
contestant chooses door number 2. The host opens door number 1, with
no prize revealed. The host then offers a second chance. 2 questions
remain .. has the ratio of chance changed from 1:3 to 1:2 AND should
the contestant change their selection to door 3?

Arguments exist that the ratio is unchanged given the selection, yet
logic suggests that the contestant should select door 3. Psychology
however hinders the thought process and fear of the failure of loss
through changing one's mind.

Any thoughts to differ?

I am no statistician but think that each participant has the same odds.
The trouble is that if they are drawing in turn, once someone has
drawn #1, all the subsquent participants will know that they cannot
draw that number and feel that they are at a disadvantage.
I would too, despite my thinking the system is fair.

IF you mark the balls with a code, maybe using four colors and four symbols 
(red, green, blue, black; square, circle, triangle, star), no one will
know what his/her ball represents, until you uncover the chart that
gives a number to each thusly coded ball.  The chart, covered on an
easel, but present from the beginning, will assure that there can have
been no manipulation.

This will keep the result a surprise till the last person has drawn.

Actually, your attempt to select a random order for the persons to
draw is superfluous, indeed, may by itself cause distrust.

Again, I am no expert, but I would accept the system.
(I once lost out on a drawing for a round-the-world flight that was
chosen by ticket number and color (red or blue).  I had the right
number, but then the wrong color was drawn.  Arghh!  We already had
plans to fly to Australia the next year!)

I concur with Myoarin. The order of persons is superfluous .. at
first. However, having spent a sleepless night upon your question, I
would consider taking the last choice, given the opportunity. Opposed
to my first view, I feel that the odds increase in favour of the
person, further down the ladder of ball plucker. I should have
realised this earlier. First choice odds are 1:16 in drawing the
winning ball. Upon drawing, the selection has now reduced to 15
possibilities. The human nature bit comes in, given that the second
drawer has "more chances" of selection as opposed to more odds stacked
against him as compared to the last, who has a chance of 1:1 .. a
given.

Again, given the choice, I would opt for last.

An interesting question.