We have a game and I can place my character in one of 4 spots, each corresponding to 1 tile (so we have spots 1 through 4 and we have 4 tiles numbered 1 through 4). We will randomly draw one, put it back and then randomly draw again. If the tile drawn corresponds to the spot where my character is, I get a game penalty.
I know that the probability of drawing tile 1 on both draws is 1/4 times 1/4 for 1/16 (6.25%). I also know that the draws, because they are random, are independent and that "history" won't matter for the 2nd draw.
Let's say that I placed my character in spot 1 and then drew tile 1 with the first draw - I got penalized for that. Now I need to place my character again in a spot and I chose to place it in spot 1 again - I know that the probability that tile 1 will come up twice is only 6.25% but my question is - for this second placement, does it matter where I place my character? The fact that tile 1 was selected first does not mean that I get a better probability of being "safe" with my character if I select tile 1 again for the 2nd draw, correct?
This seems pretty obvious but I want to be sure because of e.g. the Monty Hall problem and the fact that, in that example, switching doors actually lands you a better probability vs. keeping your initial choice.