D&D 5E Title / Subject - or probabilities are hard

[CapnZapp]

In my ToA campaign, I've reused the guts of the AL adventure Over the Edge to add some needed initial motivation for striking out "randomly" in the jungle - in my scenario the party is looking for some way to get rid of the heavy armor penalties, and Pozanna tells the party a gold statue they can trade for a solution is in one out of five sites in exchange for them mapping the sites.

They only get two (random) site coordinates, because Pozanna says other adventurers have already been given the other site locations. What they don't know is that both these other parties will have failed and died on site (somewhat like the original story). Only if they haven't found the gold statue after four sites will the fifth and final site NPC party emerge victorious, and the heroes lose out.

Anyway. Instead of already from the start rolling 1d5 to place the gold statue, I thought I make the rolls as I go along. I thought it would be boring to know already from the start if the heroes' search would be futile, and to increase the odds by giving the party "first shot" at rolling for each site.

That is, I would instead roll 1d5 when the party is visiting the first site, a 1 means the statue is there. If not, then they need to roll 1 on 1d4 when visiting the second site, and so on.

It turns out I needn't have bothered - the probabilities are exactly the same.

Probability using method #1: 20% the gold statue is in any given site, and thus it's 20% it'll be in the last site they look (or won't look as it were).

Probability using method #2: the probability it hasn't been found after visiting four sites is (1-1/5)*(1-1/4)*(1-1/3)*(1-1/2) which is... drum-roll... 20%!

So, well, anyway... just wanted to share that bit of insight.

 

[Li Shenron]

If you write the partial fractions such as (1-1/5) = 4/5, then it becomes more evident:

(1-1/5)*(1-1/4)*(1-1/3)*(1-1/2) = 4/5 * 3/4 * 2/3 * 1/2 = 1/5

It's the same for any amount of rolls:

(1-1/N)*...*(1-1/2) = (N-1)/N * (N-2)/(N-1) * ... * 1/2 = 1/N

Since each factor has the same denominator as the numerator of the previous factor, only the first denumerator (N) and the last numerator (1) remain.

 

[Schmoe]

That actually seems like a good thing! And you still have the excitement as DM of not knowing where it is until they find it.

 

[Ancalagon]

Should you roll though, or should you decide based on what is more fun?

 

[Harzel]

Wow, for me that result is quite counter-intuitive - very interesting.

It seems a shame that the PCs decisions about which site to go to next don't make a difference; in fact, whether they have a chance of success is totally out of their control. Have you considered trying to think of a way to allow them to increase their odds, perhaps at a greater risk of some kind?

 

[CapnZapp]

Wow, for me that result is quite counter-intuitive - very interesting.

It seems a shame that the PCs decisions about which site to go to next don't make a difference; in fact, whether they have a chance of success is totally out of their control. Have you considered trying to think of a way to allow them to increase their odds, perhaps at a greater risk of some kind?

I really haven't come any farther along in my thoughts process, but you're right.

Let me start out by stating the "best" outcome: since the module does detail all five locations, the ideal is if they have to visit a good number of them. But of course the best thing is if they only miss out on their prize if they make a mistake, or skip something. (Rather than just making a bad roll, that is)

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[CapnZapp]

One thought is "what if the other adventurers they find dead have already been to one other site?"

There are after all divination spells, speak with dead, and so on in the game.

Now, learning that site N conclusively does not have the gold idol is less fun (even as it increases their odds).

Perhaps they could learn one of the sites they thought was explored by others really weren't, thus changing the "bad" outcome "after searching four sites, another party triumphantly returns"...?

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[CapnZapp]

Should you roll though, or should you decide based on what is more fun?

Well, there is value in giving the players the "sandbox experience", where they can plainly see that the DM is not "stacking the deck" as it were.

Some groups might not like it, others may not care. But you have to try it to know for sure ☺

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[Ovinomancer]

Wow, for me that result is quite counter-intuitive - very interesting.

It seems a shame that the PCs decisions about which site to go to next don't make a difference; in fact, whether they have a chance of success is totally out of their control. Have you considered trying to think of a way to allow them to increase their odds, perhaps at a greater risk of some kind?

If you look at it as figuring the odds of not being successful when picking 4 out of 5, does it matter if you pick all four at once or one at a time? If no additional information is provided, the probabilities don't change.

 

[Harzel]

If you look at it as figuring the odds of not being successful when picking 4 out of 5, does it matter if you pick all four at once or one at a time? If no additional information is provided, the probabilities don't change.

Oh, heh. I had to go back and forth among the OP, my response, and your response to understand what you meant. Yeah, my post wasn't real clear about what situations I was implicitly comparing. My first comment (counter-intuitive) referred to the fact that the two methods the OP discussed for determining the location of the statue were equivalent.

But the second comment was meant to contrast the general property of both those methods (and a lot of others one could think of) - that the PCs' decisions did not affect the probability of their success - with a situation in which the PCs's decisions could affect their probability of success. As you have implied, that would seem to require giving the PCs more information on which to base their decision. I, perhaps unhelpfully, left the notion completely abstract, not attempting to describe a method that would allow the PCs' decisions to affect their probability of success. At minimum I guess I should have included a footnote that said, "I have discovered a truly marvelous way to do this, which this post is too small to contain."

Anyway, such methods clearly exist, since you could simply tell the PCs where the statue is. That by itself doesn't seem so interesting, but it could be embellished by providing supplementary goals, the pursuit of which might provide an additional reward, but decrease their likelihood of recovering the statue. Or there could be alternative paths that presented trade-offs, decreasing their likelihood of success in one way, while increasing it in another.

Alternatively, there could be clues that would narrow the scope of their search without saying where the statue was exactly, obtaining said clues itself perhaps imposing some cost.

So maybe that clarifies a bit what I was thinking about.