Odds of first draw vs last draw

Question

I have debated with myself for a while as to whether, when doing a random draw for multiple prizes, should the 'major' prize be drawn first or last? I would (as a consumer) expect that the odds or chances of winning the major prize should be smaller than than the minor prizes. So lets say there are 1000 entries and 1 major prize and 2 minor prizes. Am i correct to say that if the major prize was drawn first the odds for any one entry would be 1 in 1000 of winning, but if it was drawn third (and winning tickets were not returned to the draw) it would be 1 in 1000 multiplied by 1 in 999 multiplied by 1 in 998? ie much lower odds.

Answer 1

No.

All this does is ensure that the person who won the first minor prize and the person who won the second minor prize cannot win the major prize.

Answer 2

If the major prize is given to one of $1000$ people uniformly randomly, then the probability of any one person receiving the prize is, as you correctly stated, $$\frac{1}{1000}$$

If the major prize is given to one person of the $998$ remaining people randomly after two other prizes are given randomly, then the probability of any one of the original $1000$ people receiving the prize is still

$$\frac{999}{1000}\cdot\frac{998}{999}\cdot\frac{1}{998} = \frac{1}{1000}$$

This calculation is the probability that a person is not given the first minor prize, then not given the second minor prize, and finally is given the major prize.

Intuitively, it makes sense that it should not matter in which order the prizes are given.