Probability of choosing the correct stick out of a hundred. Challenge from reality show.

So I was watching the amazing race last night and they had a mission in which the contestants had to eat from a bin with 100 popsicles where only one of those popsicles had a writing on its stick containing the clue.

Immediately I thought well of course choosing the correct stick is 1 in a 100.
So taking the correct stick on the first try probability is $\frac{1}{100}$. Then on the second attempt it should be $\frac{1}{99}$ and so on. Multiplying these results give a huge number and so it seems that the more times you try the probability of getting the correct stick decreases.
while it seems that the more times you try it more probable for you to get the correct stick.

So how do you calculate the probability of getting the correct one first try? the second? What about last? I mean the probability of trying 100 times to get the correct stick?

Thanks.

The probability of getting the first wrong is $\dfrac{99}{100}$. The probability of getting the second right given the first is wrong is wrong $\dfrac{1}{99}$; the probability of getting the second wrong given that the first is wrong $\dfrac{98}{99}$. And this pattern continues.
Let's work out the probability of getting the correct one on the fourth try: it is the probability of getting the first three wrong $\dfrac{99}{100}\times\dfrac{98}{99}\times\dfrac{97}{98}$ times the probability of getting the fourth correct given the first three were wrong $\dfrac{1}{97}$. It should be obvious that the answer is $\dfrac{1}{100}$.
It will still be $\dfrac{1}{100}$, no matter which position you are considering. This should not be a surprise as each position of the stick is equally likely.