Dice notation

40d20

40d20
40 to 800
mean 420
mode 420

Rolling 40d20 gives a total between 40 and 800, with a mean of 420 and a standard deviation of 36.4692. There are 10,995,116,277,760,000,000,000,000,000,000,000,000,000,000,000,000,000 equally likely ordered outcomes, and the most likely total is 420 at 1.09%.

The roll

Notation40d20

Dice40

Faces per die20

Minimum total40all 40 dice showing 1

Maximum total800all 40 dice showing 20

Distinct totals761

Mean420n(f + 1)/2 + modifier, exact

Variance1,330n(f² − 1)/12, exact

Standard deviation36.46916506≈

Counting the outcomes

Ordered outcomes10,995,116,277,760,000,000,000,000,000,000,000,000,000,000,000,000,00020^40, every die distinguishable

Most likely total420

Chance of that total1.089786%≈

Ways to roll it119,823,272,437,917,606,310,254,954,069,165,933,505,260,103,622,960exact count

Chance of the minimum9.09494702 × 10^-51%≈1 in 10,995,116,277,760,000,000,000,000,000,000,000,000,000,000,000,000,000

Chance of the maximum9.09494702 × 10^-51%≈1 in 10,995,116,277,760,000,000,000,000,000,000,000,000,000,000,000,000,000

Shape of the distribution

761 distinct totals, too many to list individually.

1st percentile335≈1.006% of rolls are this or lower

5th percentile360≈5.146% of rolls are this or lower

25th percentile395≈25.154% of rolls are this or lower

50th percentile420≈50.545% of rolls are this or lower

75th percentile445≈75.711% of rolls are this or lower

95th percentile480≈95.139% of rolls are this or lower

99th percentile505≈99.067% of rolls are this or lower

Chance of rolling at least

116100%≈

230100%≈

42050.5449%≈

6100%≈

7241.99976593 × 10^-19%≈

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