Probability of Two Independent Events: Two Rolls of a Fair Die

This video explains how to determine the probability that two independent events both occur.

[loganthorne]

i would really appreciate your help understanding this.
i play card tournaments where the normal way of determining who goes first is a high roll on a 6 sided die
i prefer odds or even since its a 50:50 outcome and cant have a repeat due to a tie.
i understand that both of us have a chance to roll any given number is the same 1/6
what i dont like is giving my opponent the opportunity to have a higher roll in the first place
if they roll a 6 i then have to roll a 6 to even tie much less win.
everything im reading says they are the same odd/even or high roll and i just don't see how given in a high roll situation if the opponent is lucky enough to roll a 5 or 6 for example the amount of numbers that wins against that is less than half the numbers able to be rolled.
if you could respond id greatly appreciate it