we will consider 1,2,3,4 and 2,1,4,3 as different for the sake of simplicity
with a 1/6 chance of getting any particular side from one die, 1/36 chance from getting any particular 2, 1/216 for 3, and 1/1296 for 4 dice, you have very low odds for getting just what you want. lets start with 2 fair coins. a 1/4 chance for all heads, and 1/4 for all tails. this still means the chance of getting more heads is 1/4 because only one combo has more heads than tails
back to the dice. we will start with 2
with 2 dice, there are only 6 combos with ones and 5 of them have more ones than sixes
this puts it at a 5/36 chance to have more ones than sixes from a roll.
with all four dice its harder to calculate because you have to factor in the amount of sixes more carefully. you have to remove all the ones with at least two sixes because thats half of the dice. that rules out about 5 of the 216 options with ones. three sixes is bad too. 4 more gone. one six is okay, but if theres only a single one with it, its no good. more factors rule out more options, and i cant type long enough to find them all. hope this helps you narrow it down
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