I'm really trying to work this out for 5 six-sided dice (3 sides marked "YES" and 3 sides marked "NO") but I'm imagining that amounts to the same thing as a coin flip. I'm not well-versed in probability so please correct me if that's a bad assumption.
Odds on a fair coin flip are 50/50 heads/tails (50/50 YES/NO on a custom die).
If you make exactly 5 flips of the coin (roll 5 custom dice), counting tails, are the odds that the coin will come up more tails than heads still 50/50?
The sequence of the tails doesn't matter, just the total occurrences (this maps to more "YES"es than "NO"s on the dice).
How does increasing the number of flips (always stopping at 5 flips) change the probability of more of one face than the other?
If you roll your fair die (or flip a fair coin) any number of times the probability of more tails than heads must be the same as more heads than tails (by symmetry). If you flip an odd number of times you can't have the same number of heads and tails, so the probability that one is greater than the other is $1/2$ - there are two equally likely cases.