What is the average of rolling two dice and only taking the value of the lower dice roll? My question is related to this question, with the exception that I'm looking for the average when taking only the lower of two dice rolls.
My question is:

What is the average of rolling two dice and only taking the value of the lower dice roll?

This formula is used for the "take higher roll" case:
$$
E[X] = \sum_{x=1}^6\frac{2(x-1)+1}{36}x = \frac{1}{36}\sum_{x=1}^6(2x^2 - x) = \frac{161}{36} \approx 4.47
$$
...and includes the term "$2(x−1)+1$". Could someone please explain how that term would have to be changed so that it applies for "take lower roll"? From the original example, I'm somewhat confused where "$x-1$" comes from, although it is mentioned in the comments.
PS: I would have written this question as a comment to the original question, but I lack the points to be allowed to write comments.

Here's your payoff matrix:
{{1, 1, 1, 1, 1, 1}, 
 {1, 2, 2, 2, 2, 2}, 
 {1, 2, 3, 3, 3, 3}, 
 {1, 2, 3, 4, 4, 4}, 
 {1, 2, 3, 4, 5, 5}, 
 {1, 2, 3, 4, 5, 6}}

Sum these elements and divide by $36$.
