# Reroll & take the 2nd vs. Reroll & take the best (or worst) of two: Expected result...

### Pathfinder First Edition General Discussion

Hi all,
I was thinking about the expected values possible for the two option above and my conclusion was this:

For any use that would be reasonable the expected value would be the same BUT the first option is more versatile.

The second is useful EITHER as advantage (best) or hindrance (worst) but not for both.

The first leaves options for both since you have to state the reroll after you are told the result of the dice and you don't ask reroll if seems favourable (i.e., 10- on enemy roll and 11+ on your roll on a d20)...so in fact you are taking an "unacceptable" result as second only if it rolls worse than the "acceptable" result...

Is this thought wrong in your opinion?
Which of the two reroll would lead to better results in your opinion?

Skarm

Without running the math, the 2nd option will always be better to simulate an advantage or disadvantage since you will never "screw yourself" with the second roll.

The first option however, will nearly always "screw you" if you re-roll after achieving a roll that is already very close to the desired result.

For example. If I'm making a roll and for this roll a 20 is the ideal result and I roll a 19. I only have a 1 in 20 chance of rolling something better than what I rolled. Conversely I have an 18 in 20 chance of rolling something worse.

On the other hand if I follow the 2nd method and again roll a 19. It's to my advantage to always roll again since I have a 1 in 20 chance of rolling something better and a 0% chance of getting stuck with something worse than my 19.

The first option represents a certain amount of risk whereas the 2nd option represents no risk. IOW the 2nd one is automatically advantage or disadvantage but the 1st one carries a risk of not generating the desired outcome.

reroll and take the 2nd will be worse then reroll and pick the best.

if you then reroll and roll a 5

with the first method, you are looked into the 5. But with the 2nd one you can pick the best of the 2.

A more interesting question is.

if you have a daily limit on the reroll like 3/day. so you can't do it always.

What do you prefer/what is best?

have the ability to reroll after you have seen the result of the die.

or roll 2 dice from the beginning.

in this case, I would like the 1st option more so it's less of a chance that the ability is wasted.

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Let's run the math.

1d20 has the probability distibution of 5% chance of each value. The average value is 10.5.

Roll two d20s and take the best has the probablility distribution that the chance of rolling N is (2N-1)/400. That gives 0.25% for 1, 0.75% for 2, 1.25% for 3, ..., 4.75% for 10, 5.25% for 11, ..., 9.25% for 19, and 9.75% for 20. The average value is 13.825.

Roll two d20s and take the worst has the probablility distribution that the chance of rolling N is (41-2N)/400. That gives 9.75% for 1, 9.25% for 2, 8.75% for 3, ..., 5.25% for 10, 4.75% for 11, ..., % for 19, and 0.25% for 20. The average value is 7.175.

For roll 1d20 with an optional reroll and taking the reroll, the probablity distribution depends the circumstances under which the player rerolls.

For example, rerolling on an 8 or less gives probability distribution of 2% for each value from 1 through 8 and 7% chance for each value from 9 through 20. The average value is 12.9.

Rerolling on a 9 or less gives a probability distribution of 2.25% for each value from 1 through 9 and 7.25% chance for each value from 10 through 20. The average value is 12.975.

Rerolling on a 10 or less gives a probability distribution of 2.5% chance for each value from 1 through 10 and a 7.5% chance for each value from 11 through 20. The average value is 13.

Rerolling on a 11 or less gives a probability distribution of 2.75% chance for each value from 1 through 11 and a 7.75% chance for each value from 12 through 20. The average value is 12.975.

Rerolling on a 12 or less gives a probability distribution of 3% chance for each value from 1 through 12 and a 8% chance for each value from 13 through 20. The average value is 12.9.

Mathmuse wrote:

Let's run the math.

[...]

Interesting that the expected value peaks when rerolling half the time (10 or less).

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All else being equal, better-of-two is always a better choice.

There is some value in abilities that allow you to choose to reroll rather than roll twice in the first place--the former often let you choose to expend your resource only when you think you need it (i.e., after seeing the die roll), while the latter usually requires you to choose to expend the resource ahead of time. That's the tradeoff.

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One thing to keep in mind is that you often have a good idea whether or not your roll is good enough for your purposes. If you know that a lower roll won't make your situation any worse, you might want to consider rerolling despite the odds; similarly, if you know that a higher roll won't make your situation any better, you should not take the reroll no matter how favorable the odds are for a higher roll.

Well the first option is the monty hall problem. Will rolling a second time be better for me? In this case there is a little more data thanks to the numbers on the die. The lower the number, the better it is to risk the reroll. However; this is only really a valid arguement at 10 or less. At 10 there is a 50% chance the second roll is better. this progresses up to a 95% chance the second roll would be better if the first roll is 1.

Here are the exact percentage chances of the second method. I'll try and figure out those of the first method, just give me some time.

20-> 9.75%
19-> 9.25%
18-> 8.75%
17-> 8.25%
16-> 7.75%
15-> 7.25%
14-> 6.75%
13-> 6.25%
12-> 5.75%
11-> 5.25%
10-> 4.75%
9-> 4.25%
8-> 3.75%
7-> 3.25%
6-> 2.75%
5-> 2.25%
4-> 1.75%
3-> 1.25%
2-> 0.75%
1-> 0.3%

I'm doing this as I go, so some earlier assumptions will be discarded later. Sorry :P

the reason we can't just do a straight calculation here is because we need to know the chances someone is going to roll a second time. all our dice rolls are going to be listed as an ordered pair (1,1), (1,2)...(20,7)... etc.
So what we need to do is figure out the chances of an ordered pair from the first value(x) and second value(y)
the chance of any given y is easy; 1 in 21(the 20 options for die 2, and "dont reroll"). so the chance of any given pair is x%*(y%). But of course, y% is a constant at 1/21 so we could just as easily write it as x%/21.

I figure the chances of rerolling are either 1-(x/21) or (((x/21)^-1)-1)/21 whether we want a linear or curve approach. The purpose of the curve would be to attempt to simulate human psychology. Where we're less willing to reroll if we roll a higher number

so the linear approach gives the equation (x-(x^2/21))/21
and the curve gives us a long equation I don't feel like typing out that becomes linear and minimizes the chance of a 20 overall

OR there is the option of setting yourself a rule; if the roll 1 is less than a particular number; reroll. If that's the case, pick a number (x) on the following list. Any number less than that uses the smaller percentage, and the number and any higher use the larger percentage:
2=0.25%, 5.25%
3=0.5%, 5.5%
4=0.75%, 5.75%
5=1%, 6%
6=1.25%, 6.25%
7=1.5%, 6.5%
8=1.75%, 6.75%
9=2%, 7%
10=2.25%, 7.25%
11=2.5%, 7.5%
12=2.75%, 7.75%
13=3%, 8%
14=3.25%, 8.25%
15=3.5%, 8.5%
16=3.75%, 8.75%
17=4%, 9%
18=4.25%, 9.25%
19=4.5%, 9.5%
20=4.75%, 9.75%
to put more generally; the number chosen and those greater than it have a (10-(21-x)/4)% chance. While those lower have a (5-(21-x)/4)% chance.

So it depends on what you use to determine when you roll the second die. I hope this helps and noone yells at me for bad math XD

Thank you guys! :)

I have tried to make some calculations but I was still perplexed:

I have assumed that on the "roll again and take the best" (assuming to try to roll higher), I'd have an expected result around 17 since, in fact, I am adding the expected value of an expected second roll before calculating the expected value of the 1st.

"roll again and take the second" instead would (also assuming to try to roll higher) have lead to an expected result of 12-13...

However, I think the reality is that with "roll again and take the second" one would realistically take a reroll only when there is a fair chance to have the second roll higher...so probably almost none would roll for 11+ results of 1st roll...and someone very cautious might consider not re-rolling for 6+ results...because, in fact, the chance for the second reroll to be better are quite high...

However in both methods the expected value was around 12-13!

I was perplexed about the expected values of such rolls, but I have to agree with all of you:

"Roll again and take the best" is almost an auto-win choice, while "roll again and take the second" is more risky and less beneficial.

However since I was making this thought for the Dual-Cursed Oracle who gets Misfortune usable any time but at most 1/day per person...

I still think it is a powerful ability...

In fact I was planning to use it by multiclassing as Poisoner Rogue and I think this might give (per opponent) on average that +3 DC on poison saves which can make the difference...
Skarm

When it comes to statistics you always want to know everything you can because it's one of the few areas of mathematics where getting more information changes the answer. This is why it's sometimes said to not be a "real" math discipline.

So, in answering your original question it matters under what conditions a re-roll is occurring.

Rolling one die and then re-rolling only when the 1st die is below a certain value is quite different (statistically), than rolling two dice simultaneously and taking the better of the two values.

If you are rolling and taking the 2nd roll. It's a straight increase. If you rolled a 5 you have a 75% chance to roll something better if you rolled a 15 you have a 25% chance to roll something better. So, depending on where your limit is you know what your odds are of rolling something better. However, what you rolled before has no effect on your next roll and on a fair d20 the average value is 10.5.

However, rolling two and taking the best (using a simple estimation)
you have a 93% chance to roll better than a 5
you have a 74% chance to roll better than a 10
you have a 43% chance to roll better than a 15