# Calculating probability with a multiple of the same number on a die **URL:** \ **Category:** Factual Questions\ **Created:** [November 8, 2018, 5:38pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203 "2018-11-08T17:38:36Z")\ **Posts on this page:** 10\ **Page:** 1
**Author:** ![asterion](https://avatars.discourse-cdn.com/v4/letter/a/4da419/32.png) [@asterion](https://boards.straightdope.com/u/asterion)\ **Post date:** [November 8, 2018, 5:38pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/1 "2018-11-08T17:38:36Z")
I’m trying to figure out whether I want a D&D character of mine to take a feat that turns all 1s into 2s. It’s most relevant on casting fireball, which does 8d6 damage. The calculation is normally trivial, as the average for that roll is 28. However, if I treat 1s as 2s, that means the odds of rolling a 2 is 1:3, while the odds of 3 to 6 are 1:6 each. My die roller stats calculator can be set to a minimum value. As expected, setting the minimum roll to 2 gives the proper distribution. A 2d6 roll gives the same probability distribution as a fair roll for all totals except 4, which is doubled as expected. The average increases from 7 to 7.11. Calculating out to 8d6, setting a minimum result of 16, the average is still only 28.01. This seems intuitively lower than it should be, but that doesn’t mean I’m right. Which is why I’m asking here. So, three questions. First, am I doing the calculation right (or setting the software correctly)? Second, does this mean that on average the gain really is negligible and with enough trials the feat is worthless? (If so, sneaky, sneaky game designers). And third, as the number of possible trials is much less due to the rules of the game, is the shift to doubling the minimum damage worth it, as it only reduces the odds of something less than 16 by less than half a percent? ---
**Author:** ![Dead\_Cat](https://avatars.discourse-cdn.com/v4/letter/d/9fc29f/32.png) [@Dead\_Cat](https://boards.straightdope.com/u/Dead_Cat)\ **Post date:** [November 8, 2018, 6:04pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/2 "2018-11-08T18:04:35Z")
I don’t know about the calculator you refer to, but the average roll of a die with (effective) faces 2, 2, 3, 4, 5, 6 should be 3.67 (to 2 decimal places); i.e. 22/6, compared with 21/6 (3.5) for a standard d6. That scales up to 29.33 when multiplied by 8, not 28.01. So as far as I can determine, your intuition is right and something is off with the input or output of the calculator. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [November 8, 2018, 6:07pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/3 "2018-11-08T18:07:42Z")
I’m not totally following the OP, but note that a 6 sided die with two 2’s is not the same as a 5 sided die with no 1. If you just told the calculator to make the minimum 2 and the maximum 6, you’re describing the latter case rather than the former. ---
**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\ **Post date:** [November 8, 2018, 6:08pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/4 "2018-11-08T18:08:07Z")
If all you are interested in is the average, then each die will have a 1/6 chance of effectively being increased by 1. So for 8 dice, your effective increase is 8/6 or about 1.33. So an average roll of 29.33. ---
**Author:** ![asterion](https://avatars.discourse-cdn.com/v4/letter/a/4da419/32.png) [@asterion](https://boards.straightdope.com/u/asterion)\ **Post date:** [November 8, 2018, 6:23pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/5 "2018-11-08T18:23:23Z")
I thought it seemed off and I was giving a garbage input, but I couldn’t say why, so thanks to all. That said, the average increase is only a point and she’s getting plus three to totals already. So it’s an underpowered feat in this respect and I can get the same effect by taking her bonus to plus four. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 8, 2018, 6:55pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/6 "2018-11-08T18:55:17Z")
Keep in mind that that’s not the only effect of either the feat nor the ability score improvement. The main purpose of the feat is to deal with elemental resistance-- The converting 1s to 2s is just sort of a consolation prize for when you’re facing foes of normal flammability. ---
**Author:** ![DrDeth](https://avatars.discourse-cdn.com/v4/letter/d/b487fb/32.png) [@DrDeth](https://boards.straightdope.com/u/DrDeth)\ **Post date:** [November 8, 2018, 7:09pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/7 "2018-11-08T19:09:44Z")
Speaking as a veteran DM and designer, that feat is crap, actually. Now, a +1 to each die is \*very \*nice. But there is also a feat that does the same for Sneak attack, turning 1’s to 2’s and the math has been done- and the increase is tiny and not worth a feat. ---
**Author:** ![asterion](https://avatars.discourse-cdn.com/v4/letter/a/4da419/32.png) [@asterion](https://boards.straightdope.com/u/asterion)\ **Post date:** [November 8, 2018, 7:45pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/8 "2018-11-08T19:45:04Z")
That’s what I was thinking. And the overcoming resistance is a bit of an edge case when the character only has two fire spells anyway. Probably better to either take the ASI and up the save DC, or just take the other feat I’m thinking of. (A level 8 sorcerer with 82 HP just seems wrong somehow.) ---
**Author:** ![Aspidistra](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/aspidistra/32/3894_2.png) [@Aspidistra](https://boards.straightdope.com/u/Aspidistra)\ **Post date:** [November 8, 2018, 9:15pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/9 "2018-11-08T21:15:07Z")
If I understand what your die roll calculator is doing correctly, setting its min to 16 is not modelling your situation right. Yes, with all the 1s set to 2s you can never get under 16. But you ALSO can’t get patterns like '5 3 4 5 2 1 3 4" or “6 1 3 1 4 5 2 3” which have minimums far above that. Your die roll calculator is letting all those in - that’s why it’s so dubiously low. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [November 8, 2018, 11:11pm UTC](https://boards.straightdope.com/t/calculating-probability-with-a-multiple-of-the-same-number-on-a-die/824203/10 "2018-11-08T23:11:32Z")
Here’s the complete distribution… 5.95374×10^-7 x^48 + 4.76299×10^-6 x^47 + 0.0000214335 x^46 + 0.0000714449 x^45 + 0.000201236 x^44 + 0.000500114 x^43 + 0.00111692 x^42 + 0.00227195 x^41 + 0.00427657 x^40 + 0.00750648 x^39 + 0.0123409 x^38 + 0.0190806 x^37 + 0.0278897 x^36 + 0.0386088 x^35 + 0.050683 x^34 + 0.0631859 x^33 + 0.0749594 x^32 + 0.0845574 x^31 + 0.0906803 x^30 + 0.0924545 x^29 + 0.0896121 x^28 + 0.0822902 x^27 + 0.0715044 x^26 + 0.0586848 x^25 + 0.0453443 x^24 + 0.0326837 x^23 + 0.0219383 x^22 + 0.0136031 x^21 + 0.00767795 x^20 + 0.00381039 x^19 + 0.00167657 x^18 + 0.000609663 x^17 + 0.000152416 x^16 Where the coefficient of x^n is the probability of the dice summing to n after changing 1’s to 2’s. [Link](https://www.wolframalpha.com/input/?i=N%5B((2+x%5E2+%2B+x%5E3+%2B+x%5E4+%2B+x%5E5+%2B+x%5E6)%2F6)%5E8%5D)