Rolling twice

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I've seen a few things that allow the player to roll twice and take the higher result I was just wondering mathematically how does it effect the average of the rolls? I'm trying to figure out if they're worth it. For instance Ring of the Radiant Storm which allows you to roll damage twice and take either result, which obviously would be the higher one. And Danger Sense which allows you to roll initiative twice and take either.
iirc, it was +3.325 to the mean on a d20, but the most significant change will be in the expected range of your rolls (and the nice increase to crit%).

There was a fairly involved thread about the relative merits of Improved Initiative (+4) vs. Danger Sense (+3.325, but with caveats) some months ago.  If you can dig that up, it goes into the math side of things in greater detail.
anydice.com/program/b38

^it does that to it.
Depends on what dice you are rolling. The maximum or A and B is at least x is A or B is at least x. if A and B are independent (like rolling two dice should be) and each has probability p of being at least x, then the chance for the maximum to be at least x is p squared. With this you can easily do the math, but I wouldn't want to do it by hand. Try anydice.

Edit: ninja 
anydice.com/program/b38

^it does that to it.


Which gives you the following chance of rolling 'at least' the number ...
1: 100%
2: 99.75%
3: 99%
4: 97.75%
5: 96%
6: 93.75%
7: 91%
8: 87.75%
9: 84%
10: 79.75%
11: 75%
12: 69.75%
13: 64%
14: 57.75%
15: 51%
16: 43.75%
17: 36%
18: 27.75%
19: 19%
20: 9.75%
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For the Ring of the Radiant Storm specifically there's not a very good closed form solution that can handle such variable die rolls. If you just want to discuss d20s, it's much simpler.

For RotRS, the general rule of thumb is that the more dice you're rolling at once (and the smaller they are), the less useful it is, because the damage rolls will have less variance. It's usually not good for much more than +2 or so, to be honest (which is sad).
On a normal character it is equivalent to +5 to hit. The die value is 3.325 or whatever, but you have to factor in the number you are trying to hit. Are you trying to get anywhere between I think it was 8-12? +5. Since a 10 hits even-level on a normal character, and 8 hits even level on a minimally optimized character, dropping down to a 6 with CA....

The original math was worked out in the Avenger handbook thread somewhere. There was even a chart for what the effective bonus is at each value.
Depends on what dice you are rolling. The maximum or A and B is at least x is A or B is at least x. if A and B are independent (like rolling two dice should be) and each has probability p of being at least x, then the chance for the maximum to be at least x is p squared. With this you can easily do the math, but I wouldn't want to do it by hand. Try anydice.

Edit: ninja 

NO You square the chance of (failure on each dice), then that gives the overall chance of failure. Your mechanism makes it less likely you reach the minimum roll the more dice you roll.

If x is the chance of success with 1 roll and y is the number of independent die rolls, it's


Chance of at least one success = 1- (1-x)y
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