If greg rolls four fair six-sided dice, what is the probability that he rolls more 1's than 6's?

Each die has 6 faces, and the odds of a 6 or 1 showing up are 1 in 6. So, there are several possible victory conditions: 1 of the 4 dice shows a 1, and none show 6. 2 of the dice show one and one or none show 6. 3 of the dice show one, and 2 or less show 6. All 4 dice show 6. There are 1296 possible combinations of dice that could be rolled. So, the probabilities are as follows. 4 in 81 1 of the 4 dice shows a 1, and none show 6. 1 in 81 2 of the dice show one and one or none show 6. 5 in 1296 3 of the dice show one, and the 4 th die result doesn't matter, as long as it isn't 1. 1 in 1269 - All 4 dice show 6. So, there is a 43 in 648 chance of this occurring.