"GAME OVER, MAN. GAME OVER!"
Denmark Aarhus Unwashed and somewhat slightly dazed...

Some dice math advice needed.
Say you have a game where you roll a number of eight sided dice, typically 35, and you're rolling for a target number, typically 56, and each die that hits the target number or higher is a success, and you want to get as many successes as possible in a roll.
In this scenario, which is better, from a probability math perspective, and how do they rank?
Get an extra die. Get a reroll of one die. Get one auto succes, by changing a 'miss' to a 'hit'/success after your roll.

Graham Muller
South Africa Cape Town

Interesting worded problem,
So let's assume 3 dice with 5&6 being successess:
So for 1 dice 25% chance of a success
Successes: Probability 0: 42.19% 1: 42.19% 2: 14.01% 3: 1.56%
Add a dice changes this to: 0: 31.64% 1: 42.19% 2: 21.09% 3: 4.69% 4: 0.39%
Reroll one dice: 0:31.64% 1:42.19% 2:21.09% 3:5.08%
One auto success: 1:42.19% 2:42.19% 3:15.63%
One auto success is always best extra dice should be better than one reroll
EDIT: For the change from base probability  One auto success simply move all the base probabilities up 1 success and add the last two success chances together  One reroll: 2 successes = base 1 successes * 0.25 (probability of success) + base 2 successes * 0.75

Len Stemberger
United States Spring Texas

Here is my two cents, though I should warn you, I am not a mathematician. Also, I will assume a success is a 5, 6, 7,or 8 on a d8 (because this is 50% and easier).
The first two options are functionally equivalent (each gives you a chance to roll 1 die), except in the case where every die rolled was a success, then the reroll is wasted.
In every other case, you roll a d8 and hope for a success  50% chance.
So, because "Gain 1 additional die" is equal in all cases except where it is better, it is strictly better than "Gain 1 reroll". Additionally, "Gain 1 auto success" is also strictly better than "Gain 1 reroll" because it essentially rerolls a die but always succeeds.
Therefore, we can definitively say that "Gain 1 reroll" is at the bottom of the list.
Now for the harder discussion.
First, assume you will always roll at least 1 miss. If this is true, then "Gain 1 auto success" is strictly better than "Gain 1 additional die" because it does not have the 50% miss chance when rolled.
Second, assume you always roll 0 misses. If this is true, then "Gain 1 auto success" is strictly worse because it never triggers, and with "Gain 1 additional die" you have a 50% chance at another success.
So the question is, how likely are each of these options?
Again, assuming a success is 5, 6, 7, or 8, then the chance of rolling 0 misses is: # of dice Chance for 0 misses 3 1 in 8 4 1 in 16 5 1 in 32 6 1 in 64 n 1 in 2^n
Therefore (based on my success assumption and your initial statement of rolling 3  5 dice), the best case is "Gain 1 additional die" is better than "Gain 1 auto success" 12% of the time (1 in 8 chance with 3 dice).
So, I would rank them this way:
1. "Gain 1 auto success" 2. "Gain 1 additional die" 3. "Gain 1 reroll"
~Len

dennis bennett
Germany back in Frankfurt!
more of my stuff at www.dennisthebadger.com

jtspecial wrote: Some dice math advice needed.
Say you have a game where you roll a number of eight sided dice, typically 35, and you're rolling for a target number, typically 56, and each die that hits the target number or higher is a success, and you want to get as many successes as possible in a roll.
In this scenario, which is better, from a probability math perspective, and how do they rank?
Get an extra die. Get a reroll of one die. Get one auto succes, by changing a 'miss' to a 'hit'/success after your roll.
there migth eb a simpler (but more rough) solution:
disregard any of the original dice you're rolling.
the extra die always gives you an extra total chance of scoring (whatever you're trying to score) so think of it as a +1 chance at whatever you're trying to do.
A reroll also gives you a an extra chance, but you need to have failed first to be able to use it and it will never provide any realy "extra" option (the total of successful rolls will be lower than if you had an actual extra die)
An auto success will always be a score, so obviosuly superior to an extra die or reroll.
so ranked (but dependent on the value of having 1 die more or not, do you roll the extra die (roll x+1) but still only score the orginal number of dice (=x)?:
1. auto success 2. extra die 3. reroll
did that make any sense to you?

Jeremy Lennert
United States California

Other posters have already given you sound arguments about which of those bonuses is best and why.
If you'd like to see the exact probability distributions for each option, take a look at this program on anydice.com.

"GAME OVER, MAN. GAME OVER!"
Denmark Aarhus Unwashed and somewhat slightly dazed...

Thanks guys! This helps me a lot.
My intuitive hunches were correct, but thanks for the scientific clarifications!

Paul Dale
Australia Moggill Queensland
Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn
When will the die stop rolling? What face finishes showing?

An extra die is always better than a reroll.
An automatic success will be the best of the three, roughly equivalent to two dice with 5+ successes. Roughly, because two dice can produce two successes instead of one. If the success target is higher, the automatic is better.
 Pauli



For what its worth, the value of the automatic success over the extra die is dependent on the value they are trying to hit. It is worth noting that the lower the target number, the more attractive the additional die becomes.

David Gibbs
Canada Ottawa Ontario

As others have made clear, the automatic success is always better than the reroll one die case. As is the get an extra die case.
Whether the extra die or the autosuccess is better depends on two things: what is the successful roll odds, and what is the extra value of an extra success.
To put that second question in perspective, let's say you have the option of 3 dice, on autosuccess or 4 dice. With 3 dice, you can never have more than 3 successes, but with 4 dice you could have four. Do you need 4 successes to close the gate? Better go with the extra die. Do you need at least one success to close the gate? Better go with the autosuccess.
There are, of course, variances along the spectrum depending on how many successes you need, what the value of the extra success is, and what the odds are of a success. The lower the odds of a success, the more the autosuccess will tend to be better. The more successes you actually need, the more an extra die will tend to be useful.


