~snip~
Actually you are mistaken, the second formula you have given is the probability of rolling two sixes simultaneously with TWO dice, not rolling a 6 consecutively.
It's tricky to use probabilities in normal language because there are many possible ways to interpret what a person means.
Let's make it crystal clear continuing the dice analogy:
The probability of rolling a 6 with a dice is 1/6.
The probability of rolling two 6 in a row with one dice, which is the same as rolling two 6 with two dices, is 1/6 * 1/6 (independent events), which is 1/36.
Now, the probability of rolling a second 6 given that the previous roll was a 6, is still 1/6. In the end you still see two 6 rolled in a row, but the probability will be only 1/6 because the other event already happened or has a known outcome.This is different to the scenario where you still don't know the outcome of the two rolls, in which case the probability of it happening is 1/36.
This concept has even been used as a joke:
A famous statistician would never travel by airplane, because he had studied air travel and estimated the probability of there being a bomb on any given flight was 1 in a million, and he was not prepared to accept these odds.
One day a colleague met him at a conference far from home. "How did you get here, by train?"
"No, I flew"
"What about your the possibiltiy of a bomb?"
Well, I began thinking that if the odds of one bomb are 1:million, then the odds of TWO bombs are (1/1,000,000) x (1/1,000,000). This is a very, very small probability, which I can accept. So, now I bring my own bomb along!"