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

CapnZapp said:

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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Maybe I have this wrong, but can't probabilities ​of not finding be roughly described by (1-.02)^N in the first case, and chance at N * chance at last N concatenated in the second case?

A) When the probability of the idol being at each site is a flat 20%, the chance of not finding it at or before site N is

Site 1 = .8^1 = 80%
Site 2 = .8^2 = 64%
Site 3 = .8^3 = 51%
Site 4 = .8^4 = 41%
Site 5 = .8^5 = 33% which is forced to be 0%

B) When the probability of the ideal being at each site cascades, the chance of not finding it at or before site N is

Site 1 = 80%
Site 2 = 60%
Site 3 = 40%
Site 4 = 20%
Site 5 = 0%

Is that correct? The first case is like drawing from a deck where initially there are unlimited cards, and the one we want appears 20/100, until we've drawn four cards and then on our fifth draw all the cards we don't want are instantly removed. The second case is different, because the distribution in the deck is gradually refined forcing the probabilities to remain exact.

Is that right, or has my maths-fu failed me utterly? Summoning @Ovinomancer?

Edit: I see that I've actually added a case, where a roll is made at 20% chance at each site (which is different from making one roll up front so we know the idol is under one of these five cups, and the odds perforce change as each cup is lifted and shown to be empty.)
@CapnZapp so your initial two cases are actually one case described in two different ways, as far as I can tell, but what about this additional one? Any value in using it?

Edit 2: Makes me think you could use the - players choose the site they will visit next, right after they choose one of the remaining sites is revealed to be empty, players can now switch to choosing a different site out of those left if they want to - conundrum. (They should switch and choose a different one, is how it plays out.)

At a quick glance, vonklaude, I believe you're correct.

And no, I wasn't planning on giving them a 0.2^5 chance of going home with five statues



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Harzel said:

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.

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To the first, my point is that the two methods are identical: both have niave choosers picking 4 out of 5 options looking for 1 success. The only difference between them is the information known by someone not choosing.

Look at it this way: let's say you're on a ganesh where you have to pick from one of 5 doors, behind one of which is a car and behind the other 4 are goats. The host doesn't know which door has what behind it, and, obviously, you don't. Your odds of not picking the correct door are 20%. Now, let's say the producer of the gameshow knows what's what. Did your odds change?

Unless there's a way for you to gain information between choices, your odds are fixed no matter who else knows what.

To your second point, and to [MENTION=71699]vonklaude[/MENTION]'s question, if you did gain information, the odds do change. To Klaus' specific question, revealing one false site in this problem guarantees success, because you now have 4 guesses from 4 possibilities. To make the Monty Haul problem work with a chance of failure, you have to have at least 2 fewer choices than possibilities.

But, yes, if you make a choice of the five does, but before it's opened one of the other doors is opened to show a goat and then you're allowed to change your pick, you're better off picking a new door. This is because your choice of picking the correct door at first is 20%. Once Monty opens the other door to show a goat (and he has to show you a goat), the odds you picked the right door are still 20%, because you picked without the information. However, if you change your pick, the odds you picked a correct new door are now 25%, because you're picking from 4 doors instead of 5.

To the problem of the OP, though, I'm having trouble coming up with a way to provide the information after a choice for the next site is made but with enough time to change the choice.

CapnZapp said:

At a quick glance, vonklaude, I believe you're correct.

And no, I wasn't planning on giving them a 0.2^5 chance of going home with five statues

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Indeed not, that could risk them not finding it at all! I was thinking more that it could be fun if the odds were not fixed, and players could interact with them as they proceed. You have given yourself a way to do that, with the two ill-fated NPC parties.

Say the PCs go to site one and as expected the idol is not there. They then set out for let's say site 2, and on the way a wounded member of an ill-fated party staggers into their path "It's not site 4 or 5" she gasps out, before frothily gurgling her last. (We need to rule out two, right, based on @Ovinomancer's post?) Now the party can continue on to site 2, or switch to site 3.

Something like that, anyway?

Once past the first site, you hear someone crashing through the jungle. It's Monta Halle, fatally wounded.

Summoning her last reserves, she asks you which of the four sites you're going to visit next. Then she tells a story about one of the other three sites, where all her companions died and there were no treasure.

Then, with her dying breath, she asks, "will you keep moving to the same site, or will you switch to one of the last two sites...??"

I can add that my party is finding ever more cool sites in the jungle, and so might forget entirely about their original reason to enter it.

Which is just as it should be!

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CapnZapp said:

Once past the first site, you hear someone crashing through the jungle. It's Monta Halle, fatally wounded.

Summoning her last reserves, she asks you which of the four sites you're going to visit next. Then she tells a story about one of the other three sites, where all her companions died and there were no treasure.

Then, with her dying breath, she asks, "will you keep moving to the same site, or will you switch to one of the last two sites...??"

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Obviously, you should switch! Your chance of being correct is only 1/4, while the other two sites have a 3/8 chance.

TwoSix said:

Obviously, you should switch! Your chance of being correct is only 1/4, while the other two sites have a 3/8 chance.

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What if to choose, I roll 1d6 assigning 1-2 to the site we were headed to, 3-4 to one of the other two sites, and 5-6 to the other?

The roll comes up 1-2.

Are my odds now 1:3 of being right, even though I'm still heading for the site I chose before Monte turned up? Are the odds for each of the two other sites now also 1:3?

clearstream said:

What if to choose, I roll 1d6 assigning 1-2 to the site we were headed to, 3-4 to one of the other two sites, and 5-6 to the other?

The roll comes up 1-2.

Are my odds now 1:3 of being right, even though I'm still heading for the site I chose before Monte turned up? Are the odds for each of the two other sites now also 1:3?

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Nope. Your dice roll added no new information to the problem, so it can't affect the odds. What CapnZapp posted before is a variation on the classic Monty Hall problem (or the game show problem).

Now, I'm making some assumptions, because I know that CapnZapp is a making a joke about the Monty Hall problem. If Monta Halle's appearance is truly random (say, a 1 in 4 chance that she appeared and told the party that their current destination is wrong), than my statement above would be wrong. The Monty Hall problem only applies if the outside participant (Monta Halle, in this example) is giving information based on knowledge of the overall state of the sites.

Again, I think this is true...I'm going of what I remember of the problem from reading something about it 20 some odd years ago.

Yeah, her name was no accident

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