If I have one die and I'm rolling the dice $6$ times. What is the probability that in the all $6$ times the result will be the same?
I know that the probability for each number in $6$ sides dice is $\frac{1}{6}$
If I want the result to be 2 in all times, the probability is $(\frac{1}{6})^6$, right?
So for the $6$ numbers not needing to be $6 * (\frac{1}{6})^6$, is it right?
The other issue is, what is the probabilty that in $2$ of the $6$ rolling, it gets the same result?
For the first roll there are no restrictions. For the second, you have 5 choices. For the third, 4 and so forth. Hence, the probability of it not happening is $1\cdot \frac{5}{6}\cdot \frac{4}{6}\cdot ...\cdot \frac{1}{6} = \frac{5!}{6^{5}}$. Hence, the probability of it happening is $1 - \frac{5!}{6^{5}} = \frac{6^{5} - 5!}{6^{5}}$.