Roll & Keep: A question of probabilities

[Attilargh]

My copy of Legend of the Five Rings (3rd ed.) arrived yesterday, and I've been reading it through. I like what I've seen, but am curious: How does the Roll & Keep system affect probabilities?

For those who do not know, in L5R all rolls are done with ten-siders and are the format of XkY, where X is the number of dice rolled, k is short for "keep" and Y is the number of dice kept. The kept dice are added together and then compared to the Target Number (TN for short) set by the GM. Skill rolls are done by adding together a Trait and a Skill, and keeping a number of dice equal to the Trait. For example, when a samurai with Agility of 3 and Pole Arms of 2 wants to hit someone with a naginata, he rolls 5k3 against the target's TN to Be Hit.

Now, some situations allow a character to choose between rolling an additional die, or alternatively keeping an additional die (for example, a naginata's damage roll). Is one of the options more beneficial than the other?

I understand the basic basics of dice, but this is going quite a bit past my experience. Therefore, I would really appreciate if you tried to word your answers in layperson's terms.

[Starsinger]

Now, some situations allow a character to choose between rolling an additional die, or alternatively keeping an additional die (for example, a naginata's damage roll). Is one of the options more beneficial than the other?

Well, in general, it's beneficial to keep another dice rather than roll another die. If you are rolling 3k2 for example, each die is an equal chance to be 1-10, but your maximum is 2-20. Rolling another die gives you a better chance for that 20, but keeping another die changes your total range to 3-30. I'd personally rather keep an extra die almost 90% of the time.

[Duke of URL]

Basically, you're asking about 3k2 + 1k1 vs. 3k3?

While in theory, the third die from the first set and the single additional die are independent events with identical probabilities, this is complicated by the fact that we already know that the third die in the first roll is the worst roll of the set. I don't have the exact numbers for you, but that tells me that 3k2 + 1k1 is better than 3k3, because the 1k1 has the possibility of being better than the dice kept in the 3k2 roll.

Edit: Of course, if you're asking about 4k2 vs 3k3... without doing the math, I'd lean toward 3k3 -- min 3, max 30, avg 16.5 -- over 4k2 -- min 2, max 20, avg 11+

[Attilargh]

Basically, you're asking about 3k2 + 1k1 vs. 3k3?

Actually, I meant 4k2 vs. 3k3. Sorry for the confusion.

Ædit: Ninja'd. I've completely forgotten, how does one calculate the average on a skewed roll like 4k2?

[Duke of URL]

Actually, I meant 4k2 vs. 3k3. Sorry for the confusion.

See previous -- I had just edited it as you replied.

[Reinboom]

http://pifro.com/tempmove/xky.php
Made this for yah.

(this rolls 1000 times per combination, assuming 1d10)

--oo, this provides an interesting curve.
3k3 > 4k2, but, beyond that they get really close. Eventually, more dice rolling becomes better than keeping more.

[Attilargh]

Thanks! I'm going to bookmark that and spend some time pondering it.

[seedjar]

This is a combinatorics/statistics type of question... I believe that you would add the standard variation (the average amount that two rolls differ by) to your expected payoff (the average roll) a certain number of times according to how many dice you discarded. Not sure off the top of my head; I'd have to get out my old math books and work it out on paper. If you poke around for different combinatoric theorems involving dice, though, I'm sure you can find the right formula: mathworld.wolfram.com is a great resource.
~Joe

[Attilargh]

It seems the Playgrounders are a bunch of Scorpions. Never in my life have my edits been simu'd so many times in a single thread.

Anyway, as the probability issue is pretty much resolved (I even found a nifty probability chart) and because the thread name will probably attract a player of L5R or two: Are there any particular pitfalls in the system a rookie GM (that's me) should have to watch out for?