[Math] If you are summing the rolls of three dice, how does the expected value and variance change if you add one dice and only sum the highest three

probability

The problem is:

An alternative method to generating your character is by rolling four six sided dice
and summing only the three highest. What is the expected value and variance of this
procedure?

The previous question had the same calculation but by simply rolling 3 dice instead of 4 and summing all of them. What effect does adding the fourth dice have? Does is raise the expected value?

Best Answer

Number of cases  : 1296
Mean of score    : 12.2445987654
Variance of score: 8.1045232958

This was computed by the following Python program:

values = [1, 2, 3, 4, 5, 6]

s0 = 0
s1 = 0
s2 = 0

for a in values:
  for b in values:
    for c in values:
      for d in values:
        throws = [a, b, c, d]
        throws.sort()
        throws = throws[1:]
        score = sum(throws)
        s0 += 1
        s1 += score
        s2 += score*score

mean = float(s1)/s0
variance = float(s2)/s0 - mean*mean

print "Number of cases  :", s0
print "Mean of score    :", mean
print "Variance of score:", variance

The answer is: yes, it raises the mean (from $10.5$), but it reduces the variance (from $8.75$).

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