[Peter]

Yes Igor, the six-dice example is the same as the six-rolls example for the following reason: with the six dice, you found it easy to understand that the other dice had no influence at all on the die you were betting on. Since dice have no memory, previous rolls also have no influence on the next roll of one single die, just as the previous spin of a roulette wheel has no influence on the next spin.
Therefore, the probability of getting a specific number in the third out of six rolls is the same as the probability of rolling that number with a die that is rolled together with five different colored dice.

Peter