<s>[INST] Probability that my roll on a die will be higher than yours: Why divide by 6? I have to work out a question where on two fair, $6$ sided dice, what is the probability that the second die gives me a higher number than the first die.
So I broke the question down the long way and said "If you roll a $1$, I have a $\frac{5}{6}$ chance of beating you, if you roll a $2$, I have a $\frac{4}{6}$ chance of beating you, etc."
Then I added all these up but got the answer to be $\frac{5}{2}$.
More research showed that I was actually supposed to divide this number by $6$ to get my probability to be $\frac{15}{36} = \frac{5}{12}$, but I can't see why you divide it by $6$.
Can someone explain this to me please?
EDIT: The bit I am struggling with is the fact that why do we take the draw into consideration.
Thank you to everyone who has commented and answered, I now get that the division by 6 is because we include the probability of drawing. But in my specific question, there was nothing about a tie, I simply have to beat you, or I lose.
So why do we still include the possibility of a draw, when that's not part of the game?
 [/INST] Okay:
If the first die is a $1$ the probability of the second die being higher is $5/6$.  But that is the probability only if the first die is $1$.
If the second die is $2$ the probability of the second die being higher is $2/3$.  But again that is only the probability if the first die is $2$.
And so one.  You have six different probabilities, form $5/6$ to $0$ depending on the what the first die is.
As the first die is equally likely to be any one of those, and as it must be one and only one and never more than one of those, then the total probability of the second die being higher than the first is THE AVERAGE of all the probabilities.
To find the average probability add up the probabilities and divide by the equally likely possibilities of the first die.  i.e. $\frac 16[5/6 + 2/3 + 1/2 + 1/3 + 16 + 0] = 5/12$.
or..
(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)
(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)
(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)
(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)
(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)
(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)
Probability is: $P = 15/36$
$P = \frac {5/6+2/3+1/2 + 1/3 + 1/6 + 0}6$
$P = \frac {5+4+3+2+1+0}{36}$
$P = \frac 16*\frac 56 + \frac 16*\frac 23+\frac 16*\frac 12+\frac 16*\frac 13+\frac 16*\frac 16 + \frac 16*0$
They are all equal ways of thinking of it.
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