Sum of 4 Dies

Jacob rolls four fair six-sided dice. What is the probability that the four dice add up to 20?

Answer

When rolling four six-sided dice, there are a total of $$6^4 = 1296$$ possible outcomes.

The possible combinations that sum to 20 are:

$$ (6, 6, 6, 2) $$ : There are $$4$$ ways to arrange this combination based on the position of the $$2$$ .
$$ (6, 6, 5, 3) $$ : There are $$12$$ ways to arrange this combination, calculated as 4 times 3 =12. Here we are dealing with permutations (order of the 5 and 3 matters).
$$ (6, 6, 4, 4) $$ : There are $$6$$ ways to arrange this combination, calculated as 4 times 3 divided by 2 =6. Here we are dealing with combinations (order of the 4's does not matter).
$$ (6, 5, 5, 4) $$ : There are $$12$$ ways to arrange this combination.
$$ (5, 5, 5, 5) $$ : There is only $$1$$ way to arrange this combination since all numbers are identical.

Summing up the permutations for each combination, we have a total of:

\[ 4 + 12 + 6 + 12 + 1 = 35 \]

favorable outcomes that result in a sum of 20.

Therefore, the probability of the four dice adding up to 20 is:

P(\text{Sum is } 20) = \frac{35}{1296} \approx 0.027 \ (\text{or } 2.7\%)

Notes and comments

Comment 1: Can be pretty hard. Note that sometimes it is useful to think of 20 as 6*4=24-4, i.e. we can afford to lose 4 units at best, in this case.