Multiple dice rolling probabilities

Question

I have read some answered questions on dice rolling probabilities here but I am stuck at a point

If I am rolling 1 6-sided dice and I want to get at least 5+ the probability is $1/3 = 33.33%$

If I am rolling 2 6-sided dice and I want to get at least 5+ the probability is $1/3 + 2/3 ⋅ 1/3 = 5/9 = 55.55%$

How do I proceed with 3 or more dice if I need at least one 5+? Please explain the pattern and provide me a formula!

Thank you!

Answer 1

It's easier to calculate the probability of this not happening first.

To fail to roll 5+ once, the probability is $2/3$.

To fail to roll 5+ $n$ times in a row, the probability is $(2/3)^n$.

So the probability that you get at least one 5+ with $n$ dice is $1-(2/3)^n$.

(With $n=2$ this gives $1-4/9=5/9$, as you say.)