Just how good is rolling twice?

[logopolis]

I was flipping through some 4E skills and feats the other day, and I noticed the Agile Athlete feat (roll d20 twice for Acrobatics or Athletics checks). Clearly this was a Paragon feat because it boosted two skills, but just how much of a boost was it compared to, say, Skill Focus? Would the feat still be good if just boosted one of those skills? Being a bored math geek, I decided to crunch the numbers and find out.

As it turns out, rolling twice is really good! Assuming that a typical challenge for the players (skill checks, monster defenses, etc.) has a 40-60% chance to be overcome at equal levels, rolling twice is equivalent to about +5 on your die roll.

DC   Roll   Roll+3   Roll×2

 1   100     100      100 
 2    95     100      100
 3    90     100       99
 4    85     100       98
 5    80      95       96
 6    75      90       94
 7    70      85       91
 8    65      80       88
 9    60      75       84
10    55      70       80
11    50      65       75
12    45      60       70
13    40      55       64
14    35      50       58
15    30      45       51
16    25      40       44
17    20      35       36
18    15      30       28
19    10      25       19
20     5      20       10

DC = target number to roll
Roll = success chance with unmodified d20
Roll+3 = success chance with d20 + 3 (Skill Focus)
Roll×2 = success chance with rolling twice and taking better result
  (Agile Athlete)

 

[Aloïsius]

IIRC, I calculated that rolling twice and taking the better result gives you an average +6 or +7.
Rolling twice is really that good. This is why the paragon feat that let you roll twice for init is better than improved initiative.

 

[Mengu]

Sounds right. When I did the calculations to determine how much of a bonus Careful Attack needed, to be comparable to Twin Strike against one opponent, I came to the same conclusions. +5 is about equivalent in the 8-14 range.

 

[Eldorian]

Rolling twice squares the chance of failure. This is a good thing, since the chance of failure is between 0 and 1.

Let me introduce you to a concept called a binomial distribution.

Lets say p is a number between 0 and 1 representing the chance of success. (think of it as a percentage chance of success, where 1=1.00=100%). Then 1-p=q is the chance of failure.

The chance of getting at least one success on one roll is of course p.

The chance of getting at least one success on two rolls is p^2 + 2p(1-p)= 2p-p^2=(2-p)p=(2-(1-q))(1-q)=(1+q)(1-q)=1-q^2.

This, however, is an annoying way to think about it. Usually when determining odds with dice type questions, it's easier to calculate the odds of failure, and then success is 1- that. So when would you fail on two rolls? Two failures, i.e., q^2. So success has probability 1-q^2.

Basically, take the chance of failure, square it, and if that number is lower than the chance of failure with some sort of bonus, then a reroll is better. A reroll is only worse than a static bonus when q^2 is close to q, ie, q is big. So when it's a long shot, it might be better to have a bonus instead of a reroll.

Now, when there isn't a target number, like in initiative, then a reroll of a d20 gives expected value of 13.825, which is less than the 14.5 you get from improved initiative. However, there is less variance in the reroll, i.e., you're less likely to roll mega low or mega high when you have a reroll. This makes your initiative more predictable, which is better for you as a player (if you're designing a character to win fights. If you're designing him to have fun playing then maybe having a lot of swing factor is cool for you).

To do this is a bit harder. There are 400 possible rolls of 2d20. So basically you count how many ways of getting 1 there are, multiply that by 1, and add that to the ways of getting 2 multiplied by 2, etc.

1(1+0) +2(2+1) + 3(3+2) etc, ie the sum of k(2k-1)=2k^2 -k which requires some slightly arcane formulas to evaluate, n(n+1)(2n+1)/3 - n(n+1)/2 where n is the number of sides on the die. You divide this by the total number of possible rolls, 400, and you get the magic number 13.825. This gives the expected value of rerolling a die, picking the best of both rolls.

If there is an easier way of doing this last bit, I don't know it.

 

[Starfox]

This is why I almost always did 3 Search checks when looking for traps back in 3E. Making 3 checks gives me a 7/8 chance to roll 11+, which should discover most traps, and is MUCH faster than taking 20.

A problem is that you can't simply replace a "roll twice" effect with a +6 bonus. They do completely different things. The +6 bonus almost guarantees that you won't roll poorly, but it does not increase the maximum possible result.

 

[Plane Sailing]

A problem is that you can't simply replace a "roll twice" effect with a +6 bonus. They do completely different things. The +6 bonus almost guarantees that you won't roll poorly, but it does not increase the maximum possible result.

I'm sure you meant to say that "The reroll almost guarantees that you won't roll poorly, but it does not increase the maximum possible result.

Cheers

 

[Bagpuss]

Yeah the almost is the most important bit, I've lost count of the number of times I've used Elven Accuracy and still missed.

 

[phaed]

I just wrote a quick script to see what the average actually is for this.

Running a million iterations a couple different times comes out to an average result of 13.825, which is effectively (on average) slightly better than a +3 bonus over the 10.5 average of one d20.

So over your character's lifetime, the agile athlete feat would be slightly better than skill focus in both of those skills.

For the heck of it I also ran it with more rerolls:
1d20 avg: 10.5
Best of 2d20: 13.825
3d20: 15.49
4d20: 16.48

 

[Staffan]

I just wrote a quick script to see what the average actually is for this.

Running a million iterations a couple different times comes out to an average result of 13.825, which is effectively (on average) slightly better than a +3 bonus over the 10.5 average of one d20.

The thing is that the average skill check value isn't really what's relevant. What's relevant is your chance of success.

The table the OP posted shows that "reroll" is better than skill focus when the DC is between your skill bonus +5 and +17 (without any hypothetical skill focus bonus, of course). That is, in almost every circumstance you'd be likely to run into.

 

[Cadfan]

Yeah the almost is the most important bit, I've lost count of the number of times I've used Elven Accuracy and still missed.

Elven Accuracy is a little bit of a different ballgame, since you only use it on known misses.