Question about finding sum of 3 largest rolls of die

#1
The S sum of all 4 rolls = 14

The E(M) minimum value of 4 rolled die is 1.755

The example here my professor worked out for the Expected Sum of the 3 largest rolls = 14(sum of all 4 rolls) - 1.755(minimum value of 4 rolled die) = 12.245

I don't understand how the Expected Sum of the 3 largest rolls = the total sum - minimum value of 4 rolled die:confused:
 

Mean Joe

TS Contributor
#2
Given that the sum of all 4 rolls is 14, then the expected minimum value of 4 rolled die is 1.755. Note that the minimum value must be either 1, 2, or 3 (impossible for min value to be 4, because then the sum of all four rolls is at least 16).

So if you have the min value, then the other three rolls = the three largest rolls. And three largest + minimum = four rolls.
 

BGM

TS Contributor
#3
Or simply the unordered sum is equal to the ordered sum:

\( X_{(1)} + X_{(2)} + X_{(3)} + X_{(4)} = X_1 + X_2 + X_3 + X_4 = S \)

\( \Rightarrow E[X_{(2)} + X_{(3)} + X_{(4)}|S = 14]
= E[S|S = 14] - E[X_{(1)}|S = 14] \)