#### MrPokemans

##### New Member

- Joined
- Dec 3, 2013

- Messages
- 10

**The situation**: I have 10 standard six-sided dice and want to roll values of 4, 5 or 6. This for example represents a game with 10 Goblins attacking a Troll, needing a 4 or higher to damage it. However, the Goblins are quick to strike so they can reroll their failed rolls once. So after having rolled the 10 dice the first time I take all the failed dice (which had values 1, 2 or 3) and roll them once more in an attempt to get more 4's, 5's and 6's.

When repeating this test a thousand times I'd expect the 10 dice to have 5,0 success rolls (on average) on their first attempt because there is a 50% for a dice to roll a 4 or higher. After the rerolls, I'd expect 7,5 success rolls because the chance has increased to 75%.

**The question**: Why are the 10 dice in my Excel sheet getting only about 5,8 success rolls on average, instead of the expected 7,5? Is there a mathematical explanation for this or do my formula's contain an error?

**The formula's I'm using**:

Column A

=RANDBETWEEN(1;6)

=RANDBETWEEN(1;6)

=RANDBETWEEN(1;6)

=RANDBETWEEN(1;6)

=RANDBETWEEN(1;6)

=RANDBETWEEN(1;6)

=RANDBETWEEN(1;6)

=RANDBETWEEN(1;6)

=RANDBETWEEN(1;6)

=RANDBETWEEN(1;6)

Column B

=IF(A1<4;RANDBETWEEN(1;6);A1)

=IF(A2<4;RANDBETWEEN(1;6);A2)

=IF(A3<4;RANDBETWEEN(1;6);A3)

=IF(A4<4;RANDBETWEEN(1;6);A4)

=IF(A5<4;RANDBETWEEN(1;6);A5)

=IF(A6<4;RANDBETWEEN(1;6);A6)

=IF(A7<4;RANDBETWEEN(1;6);A7)

=IF(A8<4;RANDBETWEEN(1;6);A8)

=IF(A9<4;RANDBETWEEN(1;6);A9)

=IF(A10<4;RANDBETWEEN(1;6);A10)

Column C

=LEN(SUBSTITUTE(SUBSTITUTE(SUBSTITUTE(A1&A2&A3&A4&A5&A6&A7&A8&A9&A10;3;"");2;"");1;""))

[cells C2 to C10 are empty]

Column D

=LEN(SUBSTITUTE(SUBSTITUTE(SUBSTITUTE(B1&B2&B3&B4&B5&B6&B7&B8&B9&B10;3;"");2;"");1;""))

[cells D2 to D10 are empty]

Select cells A1 to D10 and drag the selection down to row 1.000 to copy the ten dice being rolled and rerolled a hundred times. Or drag them down to row 10.000 or 10.000 for even more iterations.

To conclude,

__if you select the entire column C, the numbers in the bottom right of he screen give an average value of between 4,8 and 5,2__. This is the expected 50% chance to roll a 4 or higher in the first attempt, and in my sheet these values are

__confirmed__.

However,

__when selecting column D, it shows an average of about 5,8 instead of the expected 7,5__. How is that possible???

Any help would be highly apreciated!

Regards, Joris