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