Say you want to compute the probability of getting a $3$.
If the result is $3$, then it is because you got $1$ on every dice, right? There can be no other way to get a $3$ as a result. So,
$$P(\text{result is $3$}) = P(\text{all dice roll $1$}) = \frac16\cdot\frac16\cdot\frac16\cdot\frac16 = \left(\frac16\right)^4$$
(I am assuming you know how to calculate the probability of a specific outcome of a single die);
How does one find $P(18)$? If you get $18$, three die rolled $6$ right? It does not matter what did the fourth die roll, because it will never be higher.
Lets say your dice are coloured, all with different colours. One red, one blue, one black and one white. Let's say the red was the one you ignored. The chances of getting $6$ in all other dice is
$$P(\text{white, black and blue rolled $6$}) = \left(\frac16\right)^3$$
But you chose a specific coloured die to be ignored. In how many ways could you make such a choice? In $4$ ways. So you should multiply that by $4$ right?
That would give $4\cdot\left(\frac16\right)^3 \approx 1.85\%$ which is more than you got. Why? Because you counted one specific event $4$ times: when all $4$ dice roll $6$. So you should subtract $3$ of those events, so as to take that into account only once:
$$4\cdot\left(\frac16\right)^3 - 3\cdot\left(\frac16\right) \approx 1.62\%$$
For other specific outcomes you think similarly. Either by counting every single way of that outcome happening, or by this method of starting by general things and then taking into account what was added or subtracted too many times.
Best Answer
There are 36 possible outcomes when throwing two dices, as there are 6 outcomes throwing only one dice. But, naming a dice $A$ and the other $B$, there are two outcomes you are looking for; $A=1$, $B=2$ and $A=2$, $B=1$.
So the probability of getting 1 and 2 by throwing two dices is $$ \frac{2}{36}=\frac{1}{18}. $$ Then, to get 1 and 2 seven times in a row, $$ \left(\frac{1}{18}\right)^7=\frac{1}{612220032}\cong1.63\cdot10^{-9}. $$ So the rate would be $612220032:1$. Still, this is 10 times bigger than the one you said of 62 million. I searched on the internet about this episode and I found a quote in IMDb, saying exactly what you said. I really believe that the show is wrong.