TITLE: What is the average of rolling two dice and only taking the value of the higher dice roll?
QUESTION [18 upvotes]: What is the average result of rolling two dice, and only taking the value of the higher dice roll?
To make sure the situation I am asking about is clear, here is an example: 
I roll two dice and one comes up as a four and the other a six, the result would just be six.
Would the average dice roll be the same or higher than just rolling one dice?

REPLY [13 votes]: I'll have a go and answer this the maths-lite way (though there are a number of answers with more mathematic rigor and .. dare I say it vigor posted here already).

Note that there is:

1 result with a face value 1
3 results with a face value 2,
5 results with a face value 3,
7 results with a face value 4,
9 results with a face value 5, and
11 results with a face value 6

The Average is defined to be: 
$$\text{Average} = \frac{\text{Sum of the Results}}{\text{Total number of Results}}$$
The Sum of the Results is:
$$\begin{eqnarray} 
\text{Sum} &=& (1 \times 1) + (3 \times 2) + (5 \times 3) + (7 \times 4) + (9 \times 5) + (11 \times 6)     \nonumber \\
           &=& 1 + 6 + 15 + 28 + 45 + 66 \nonumber \\
           &=& 161    \nonumber
\end{eqnarray}$$
The Total number of Results is: $ 6 \times 6 = 36$
So the Average is:
$$\text{Average} = \frac{161}{36} \approx 4.472$$