Dice Rolling Paradox?

Verdigris

First Post
For several weeks now I have been contemplating what appears to be a statistical paradox. Since my highest math skills consist of discretely copying Brian C.'s answers over his shoulder in Algebra class some yeas ago, I thought I would pose my muse to the board and see if one of you can answer it satisfactorily. Forgive the muddled language: it is a bit tricky to explain.

When D&D requires a percentile dice roll, say 25% or less, ordinarily you roll a d10 for the 10s column and a d10 for the ones column. Nothing strange about that, right? Its in the rules, right?

Well, consider this. If you roll your dice one at a time something odd appears to happen.

First, you roll the 10s column. In order to make your percentage, you have to roll a 1 or a 2. Statistically, you have a 1/5 or 20% chance of doing this.

Irrespective of whether you roll a million more dice after your first roll, the threshold you always need to pass through is 1 in 5. It must always pass through 20% on its way to whatever final roll you are trying to make. This initial chance can neither be lessened or increased by rolling the "1's" column, because the first roll is already tabulated by the time you roll the second.

Oddly, as it happens, the "1s" column appears even to *lessen* your chance of making your 25% because, if after the 10s roll (i.e. the 20% threshold), if you've gotten a "2", you've still only got a fifty-fifty chance of making your "1s" roll (i.e. 1-5 vs. 6-10).

Your 25% roll suddenly looks like 20% that only succeeds three fourths of the time.

Where did I go wrong?
 

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Verdigris said:
When D&D requires a percentile dice roll, say 25% or less, ordinarily you roll a d10 for the 10s column and a d10 for the ones column. Nothing strange about that, right? Its in the rules, right?

Well, consider this. If you roll your dice one at a time something odd appears to happen.

First, you roll the 10s column. In order to make your percentage, you have to roll a 1 or a 2. Statistically, you have a 1/5 or 20% chance of doing this.

To roll 25%, you need to roll a 0, 1 or 2 on the 10's die, and then if you roll a 2, you need to roll a 1 to 5 on the 1's die.

(Not exact, but close enough for the purposes of this thread.)
 
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As Hong pointed out you went wrong on the first step. You need a 0,1 or 2 thats a 30% chance on the first roll not 20%.

Also only half of the 2 results on the first roll will result in success so half (actually its just over half as its six values 0 to 5) of the 10% chance of rolling a 2 give 5% added to the 10% from rolling 1 and the (fractionally under, you could roll 0,0) 10% for rolling a 0 on the first dice, it all works out to 25%. There is no paradox.
 

A little more exact:

If you roll a 0 first (10 % chance), you have to follow up with a 1-9 (90 %).
If you roll a 1 first (10 %), you can roll anything on the second die (100 %).
If you roll a 2 first (10 %), you have to roll 1-5 or 0 (60 %).

Probability of success = 0.1 x 0.9 + 0.1 x 1.0 + 0.1 x 0.6 = 0.25 or 25 %.

So it all adds up correctly.
 

*Verdigris [wincing] casts Bigby's Forehead Smack on himself and quaff's a potion of "Restoration" to regain whatever points of Intelligence he obviously lost considering this nonexistant paradox*
 

Stuff like this used to really get one of my buddies upset. He swore that the odds were different if I used a d6 and a d10 to improvise when I couldn't find a d20 (d6: 1-3 = add 0 to d10 result; 4-6 = add 10 to d10 result).
 

it might not have been the statistics that bothered him as much as the lack of a d20. a long time ago I houseruled that. if the situation calls for a d20 roll then thats what you roll. if you don't have one then go down to the local gaming store and invest a buck. at least thats what I've told my players for years.
 

Verdigris said:
*Verdigris [wincing] casts Bigby's Forehead Smack on himself and quaff's a potion of "Restoration" to regain whatever points of Intelligence he obviously lost considering this nonexistant paradox*

LOL! I have to add "Bigby's Headslapping Hand" to my list of silly cantrips...
 

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