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