What are the chances to have three equal numbers on dice, three times in a row?

Question

So, my friends and I were playing a game with dice, and we used to get three equal numbers three times in a row. So we are wondering, what are the probabilities for something like that to occur?

Answer 1

For three dice to land on the same number, whatever the first die is, the second and third must have its value. The probability of this happening is $\frac16$ for the second die, times $\frac16$ for the third die.

Now, the probability of this event occurring thrice is simply the probability of it occurring once, cubed. This gives a final answer of $$\left(\frac16\times\frac16\right)^3=\frac1{6^6}\approx0.0021433\%.$$