Dice notation

31d20

31d20
31 to 620
mean 325.5

Rolling 31d20 gives a total between 31 and 620, with a mean of 325.5 and a standard deviation of 32.1053. There are 21,474,836,480,000,000,000,000,000,000,000,000,000,000 equally likely ordered outcomes, and the most likely total is 325 or 326 at 1.236%.

The roll

Notation31d20

Dice31

Faces per die20

Minimum total31all 31 dice showing 1

Maximum total620all 31 dice showing 20

Distinct totals590

Mean325.5n(f + 1)/2 + modifier, exact

Variance1,030.75n(f² − 1)/12, exact

Standard deviation32.10529551≈

Counting the outcomes

Ordered outcomes21,474,836,480,000,000,000,000,000,000,000,000,000,00020^31, every die distinguishable

Most likely total325 and 326two totals share the peak

Chance of that total1.236402%≈

Ways to roll it265,515,250,781,181,106,065,702,854,328,695,221,376exact count

Chance of the minimum4.65661287 × 10^-39%≈1 in 21,474,836,480,000,000,000,000,000,000,000,000,000,000

Chance of the maximum4.65661287 × 10^-39%≈1 in 21,474,836,480,000,000,000,000,000,000,000,000,000,000

Shape of the distribution

590 distinct totals, too many to list individually.

1st percentile251≈1.034% of rolls are this or lower

5th percentile273≈5.275% of rolls are this or lower

25th percentile304≈25.74% of rolls are this or lower

50th percentile326≈51.236% of rolls are this or lower

75th percentile347≈75.252% of rolls are this or lower

95th percentile378≈95.054% of rolls are this or lower

99th percentile400≈99.051% of rolls are this or lower

Chance of rolling at least

90100%≈

17899.9999%≈

32650%≈

4730.0001%≈

5616.05650302 × 10^-15%≈

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