Forgive me, I read through that entire sticky post at the top, and at the end, if anything, I was more confused. So I'll spell everything out here, and if one of you fine talented folk can tell me if I'm on the right track or not you will make my day!

I have ten cards turned face down. One has a square on it, the other 9 have circles.

I believe I have a 1/10 or 10% chance of drawing the square.

I believe that I have a 9/10 or 90% chance of drawing a circle card.

If, after each draw, the cards are shuffled, so that it is an entirely random process once again, entirely separate from the initial draw, what are the odds that after two draws (at least) one of them would have been a square?

1/10 * 1/10 = 1/100? But that's not right. I know the odds are higher than that.

9/10 * 9/10 = 81/100 of having drawn nothing but circles? That sounds like it could be right, and if so means that the remainder, 19% is the correct odds for (at least) 1 of my 2 cards having been a square?

9/10 * 9/10 * 9/10 = 729/1000 for 27.1% after 3.

9/10 * 9/10 * 9/10 * 9/10 * 9/10 * 9/10 * 9/10 * 9/10 * 9/10 = 387420489 / 1000000000 = 0.387420489 or ~61.25% of after 9 draws having (at least) 1 of them be a square.

9/10 * 9/10 * 9/10 * 9/10 * 9/10 * 9/10 * 9/10 * 9/10 * 9/10 * 9/10 = 3486784401 / 10000000000 = 0.3486784401 or ~65.1% after 10 draws

The first part of my question is, am I right?

The second part of my question is:

How many times would I need to draw to have a 90% chance of having drawn at least 1 square card?