Ring of Anticipation - Initiative bonus math

[odigity]

Just learned about this item:


Ring of Anticipation (DoTU pg 100) 6000gp CL7 "When making initiative checks, you can roll twice and take the better result."

I was curious, so I did some math to figure out how much better it would make things.

{table=head]Roll|1d20|2d20 (best of)

1|5%|0.25%

2|5%|0.75%

3|5%|1.25%

4|5%|1.75%

5|5%|2.25%

6|5%|2.75%

7|5%|3.25%

8|5%|3.75%

9|5%|4.25%

10|5%|4.75%

11|5%|5.25%

12|5%|5.75%

13|5%|6.25%

14|5%|6.75%

15|5%|7.25%

16|5%|7.75%

17|5%|8.25%

18|5%|8.75%

19|5%|9.25%

20|5%|9.75%

[/table]

It's hard to process at that level of detail, so I then summed it up in quarters:

{table=head]Roll|1d20|2d20 (best of)

1 - 5|25%|6.25%

6 - 10|25%|18.75%

11 - 15|25%|31.25%

16 - 20|25%|43.75%

[/table]

Then the minimum top range to get >= 50%:

{table=head]Roll|1d20|2d20 (best of)

15 - 20|30%|51.00%

[/table]

Finally, I multiplied each of the 20 roll results by their percentage chance, summed those totals, and divided that total by 100 to get the average roll...

{table=head]1d20|2d20 (best of)

10.5|13.8225

[/table]

...which was surprising to me. I figured if the 15-20 range is 51% likely, then the average roll should be just a hair above 15. Not sure if I've done anything wrong here, but the result doesn't make sense to me.

Assuming I'm right, the Ring of Anticipation is essentially equivalent to an untyped +3.3225 bonus.

[Draz74]

Thing is, 2d20b1 doesn't have the same shape of probability curve as 1d20, so the change can't be summarized just by approximating it as a bonus to the same old roll.

In fact, rather than the average (mean) that you calculated, you'd get a better idea for how advantageous the mechanic is if you calculated the median. Which IIRC comes out to be 14. So if anything, it's closer to a +4 bonus than a +3.2 bonus.

[Jeff the Green]

Right. 2d20b1 is left skewed (http://en.wikipedia.org/wiki/Skewness). The left tail is longer than the right, so the mean is influenced strongly by a few low rolls while the median isn't.

[odigity]

Well, the median would happen at the 50% mark, so just a tad above 15.

So, +3.3225 mean, slightly over +4.5 median.

I like it. (Also gives +2 Listen/Spot [competence].)

[Thurbane]

This is one of those great, "obscure" (i.e. non core, non MIC) magic items that is really, really nice (like Collar of Umbral Metamorphosis, Tooth of Savnok or the Smoking weapon enhancement).

It's like an all day long version of the Danger Sense feat. :smallbiggrin:

[odigity]

And it's not actually a bonus, so no type/stacking issues. :)

Credit goes to this thread (http://www.giantitp.com/forums/showthread.php?t=187851).