Easy Math help (for some of you)

Can someone explain how to calculate the probability of the following events?

Let's say there are 10 people and they are going to draw a card number 1 to 10. 1 wins, 2 comes in second, 3 third on so on.

Being that there is no skill involved I think it is correct to say that all players have a 1 in 10 chance or 10% chance of winning. (Or finishing in any specific spot) right?

If the same players where to repeat the same game would the odds of the same player winning be 1 in 100 or 1%? (10 * 10)

If the second time there were 20 players and 20 cards would the odds of the first player winning again 1 in 200 or .5%?

If there is a third or forth game do you just keep multiplying?

Thanks