Am I more likely to roll at least one six if I have more dice?

Hi,

I've had a debate with some friends, about whether I'm more likely to roll at least a single 6 when I have 1000 dice, than I am to roll one with only 1 dice.

I say, the more dice I have, the more likely it is to roll at least a single 6. The others try to convince me, that the possibility to roll at least one 6 is always 1/6.

Now my thoughts for this are as follows:

At first, it doesn't matter, whether I roll them one after another, or one at a time.

The chances to roll a six on the first dice that I roll a 6, is 1/6.

Now, I don't care about the others if I did roll a 6, but I do care if I didn't (which is in 5/6 of all cases)

So for the second dice roll a six, the probability is still 1/6

However, I'm wondering about the probability that the first dice does NOT show a six, but the second does, so the probability for that to happen is 5/6*1/6.

So the probability to roll at least one 6 with two dice is:

1/6 + 1/6*5/6

If I was to expand this for 1000 dice, I would end up with:

$\displaystyle $$\sum_{k=0}^{999}\left (\frac{5}{6} \right )^k \times \frac{1}{6}$$$

The probability will get ever more closely to 100%, the more dice I have, correct?