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August 14

Trying to figure out the probability that my players pass this dice roll

Hi all. I have a party of four Call of Cthulhu RPG players who are going to find a damaged radio in their next session. None of them put skillpoints into the skill needed to repair it, meaning all four of them have the default 10% chance of success. I'm trying to figure out what their odds are of success using the following assumptions:

In CoC, players roll a d100 (actually two dice, a d10 and a d-percentile which represents the 'tens' place of their roll result) for skillchecks, and their goal is to roll low. In this case, when a player attempts the check, they succeed if they roll a 01, 02, 03, 04, 05, 06, 07, 08, 09, or 10 out of the 100 sides.
Knowing my players, they're going to pair off, two of them will use the "help" action to give the other two a 'bonus die' on their attempt to repair the radio. (Update: This means the two helping do not get to make a check to fix the radio themselves.) What this means is that both attempters get to roll their d-percentile a second time and use the lower of the two values (remember, low values are better in CoC). Only one of the two attempters needs to pass the check to repair the radio.

So, assuming the party does what I expect, what are the odds they successfully roll to repair the radio? --Aabicus (talk) 20:30, 14 August 2023 (UTC)[reply]

First, isn't 00 a valid dice role? An any case, the actual mechanics of generating a 10% chance are irrelevant. Also, I don't think pairing really matters at all since the upshot is that if anyone gets a win in four tries then it's a success. It's easier to find the probability of failure. All four players would have to fail, there's a 90% chance of each player failing, so there is a .9⁴=65.61% chance that all of them will fail. That means a 34.39% chance of success. --RDBury (talk) 22:13, 14 August 2023 (UTC)[reply]

00 is considered "100" on a d100, ergo a failure in CoC rules. There's a consideration I didn't mention that explains why the four don't all do the checks individually (rolling 95-100 is a fumble, and if any of them fumble the radio would break irreparably. They've been burnt by fumbles in the past so they prefer to do the bonus dice thing which makes fumbles far less likely to roll). Which is why I'm hoping to know the odds if two of them use the bonus dice style of rolling, instead of straight 4 checks at 10% each. --Aabicus (talk) 22:22, 14 August 2023 (UTC)[reply]

Ignoring the fumble for a second, it doesn't matter. Think of it like this: If the helpers help, they get to reroll one die for their teammate: the d-percentile die. If they don't help, they get to reroll both of their teammate's dice (they actually get to roll their own dice, but there's no difference between rolling dice that belong to you and rolling dice that belong to someone else). If anything, not helping might be better. But it turns out in this specific case, the probabilities work out the same.

With fumbling, there needs to be a clarification I think. Does a teammate have to announce ahead of the roll whether they're helping? Or can they look at the result and say "I'll spend my help action to reroll the d-percentile die"? Closhund /talk/ 23:23, 14 August 2023 (UTC)[reply]

They've got to declare beforehand that they're helping a specific ally's attempt (and sacrificing their own opportunity to try by doing so), instead of taking their own attempt to fix the radio. That's a fair point, how in the end the party is still rolling 4 times, and they still only need a single one of those checks to pass. Like you mentioned, the only difference seems to be that the "4 straight checks" has four opportunities to fumble, whereas the "2 checks with bonus dice" has significantly lower odds of fumble (either attempter would need to fumble on both their regular and bonus roll.)

My brain's having trouble committing to the thought that "reroll both dice" and "reroll only the tens die" are essentially equivalent when it comes to a player's odds of passing the check, but I can kinda see why that'd be the case. Indeed, I can't think of a concrete reason why it wouldn't be, it still boils down to whether that tens die comes up 00 vs any other number.--Aabicus (talk) 23:37, 14 August 2023 (UTC)[reply]

There should be 100 possible outcomes of the two-dice roll. Are the possible outcomes 00 through 99, or 01 through 100? A two-player team fails if the first player's roll fails and then the second player's roll also fails. We may assume the rolls are independent. This means that the probability of a double failure is the product of the probabilities of the single failures. If 10 out of the 100 equally likely outcomes mean success and 90 mean failure, the probability of a single failure is 90/100 = 0.9. That of a double failure is then 0.9 × 0.9 = 0.81.. The probability of the complementary event, team success, is its complement with respect to 1, 1 − 0.81 = 0.19, or 19%. --Lambiam 22:21, 14 August 2023 (UTC)[reply]

Rolling a 00 is considered 100 on a d100, so the outcomes are 00-99 but 00 is still a fail from the perspective of the players. Does that 19% include the 'bonus dice' mechanic whereby both attempters get to reroll their d-percentile and hope for a lower digit in the tens-column of their result? --Aabicus (talk) 22:27, 14 August 2023 (UTC)[reply]

So the "fumble" mechanic makes things a bit more interesting. With taking the best of two I gather that the chance of a fumble is .05² = .0025. For each pair of rolls, the chance of a fumble or immediate loss is .05, the chance to not win otherwise is .9²-.0025=.8075 and the chance of a win is 1-.9²=.19. You only roll a second time if there is a "not win", so multiply those odd by .8075, you get .00201875 chance of fumble on the second pair of rolls, .65205625 of a not win, and 0.153425 of a win. The totals are .343425 to win, and .656575 to lose. Note that the fumble rule does not significantly change the outcome, and that makes sense considering it's so unlikely.

In the context of a game, small differences in probability aren't really noticeable. To distinguish between 34% and 35% to any statistical significance would take hundreds of repetitions, and most people aren't going to do that, and be willing to carefully record the results, unless they're being paid for the effort. Popular games tend to have complicated mechanisms built in to make find exact odds difficult to compute; people seem to prefer having some kind of suspense in the process. --RDBury (talk) 01:55, 15 August 2023 (UTC)[reply]

Thank you for breaking it down! This is pretty much exactly what I was hoping to know, they have roughly a 34% chance of fixing the radio. I'm also able to follow your train of calculation so I can apply it to situations in the future to determine roughly how likely or unlikely some certain check will be for this party. Thanks again! --Aabicus (talk) 08:09, 15 August 2023 (UTC)[reply]

It's still not clear to me. Let's call the teams team A and team B, and the players A1, A2, B1 and B2. A1 rolls both the d100 and the d%ile and gets 23, say. Does player A1 now re-roll the d%ile, and if they get 73 player A2 can now also roll both dice and so on? Or does A1, after getting 23, let A2 re-roll the d%ile in the hope of getting 03? If both A1 and A2 can roll both dice and re-roll the d%ile, the probability of one-player failure is 0.81 and of team failure is 0.3439. Also, do you want to know the probability of "at least one of teams A and B gets through" or of "A gets through"? --Lambiam 06:52, 15 August 2023 (UTC)[reply]

After rereading I now think that the "attempters" from the question are A1 and B1, and that A2 and B2 never roll the dice. The probability that an individual attempter fails is then 0.81. The probability that both attempters fail is 0.3439. This ignores fumbling. Apparently, there is one radio, and if one team breaks it, the other team does not get a chance. Can A1 roll the dice, and if getting 95, still reroll the d%ile die? Or is it then game over (as far as the radio is concerned) for everyone? --Lambiam 07:11, 15 August 2023 (UTC)[reply]

In that scenario, A1 would get to use their bonus die and reroll their d-percentile (assuming a bonus die is in play, granted from A2 in your example) and the skillcheck wouldn't be a fumble unless that bonus die also came up a fumble. If there isn't a bonus die in play (e.g. all four players chose to attempt to fix the radio, no helping), any one of them rolling a fumble breaks the radio. (And each attempter would roll one-at-a-time, not all at once, so there's no chance of "A1 fumbled, but B1 fixed the radio at the same time" sort of trouble.) --Aabicus (talk) 08:04, 15 August 2023 (UTC)[reply]

The possible outcomes for a single attempter are success, (simple) failure and fumble. The probabilities of these three outcomes are 0.190 for success, 0.756 for failure, and 0.054 for fumble. The radio gets repaired if team A succeeds, or if team A fails but does not fumble and team B succeeds. This gives a probability of repair of 0.190 + 0.756 × 0.190 = 0.33364. --Lambiam 09:06, 15 August 2023 (UTC)[reply]

Awesome, thank you so much! --Aabicus (talk) 18:47, 15 August 2023 (UTC)[reply]

In future you can note that for games like CoC, the probability distribution for the minimum of two uniform dice rolls looks like a one-sided triangular distribution, which has a S-shaped cumulative distribution (i.e., probability that an assisted check succeeds). So if you can follow along a bit on the calculations here, you could also try plotting it out on a graph with different choices for the success threshold. (For a very basic introduction to simple dice probabilities, that includes the PDF and CDF, see a 2018-05-09 blog post by Mazur.

Responding to RDBury: FYI most RPGs I've examined don't actually have complex dice mechanisms from a probability perspective -- at most they have a triangular PDF (the most curvature you can get out of two fair or Fudge dice rolls), but most use uniform distributions on top of which they add bells and whistles that are just equivalent linear shifts (see e.g. most editions of D&D for the most egregious, most popular example, that somehow makes it also the most confusing and mentally difficult). The most sophisticated popular RPG mechanic in probabilities, though still relatively easy to calculate analytically, as well as to process mentally during gameplay, are the White Wolf games. A consideration with dice mechanisms is of course always the speed and complexity of mental calculation. But from my dive into the subject of tabletop gaming, most of the change in usability comes from design and testing over probability theory. Some may even purposely design against usability -- it seems D&D designers and players embrace their illusion of mathematical complexity and the sense of exclusivity that their frustrating early mechanics brings -- and I can see the appeal from watching especially retrogamer groups. (And if actually do you know someone willing to pay for my effort on this, then please let me know.) SamuelRiv (talk) 11:49, 15 August 2023 (UTC)[reply]

Just as an aside on the gameplay issue; as a DM, your role is to keep the game going and make it fun and challenging. You can always bend the rules a bit for the sake of gameplay. Dice rolling is a useful way to add tension and randomness to a game session, and present unforeseen challenges that force you and the players to think and adapt to the situation quickly and in novel ways. Having an "out" to keep the plot moving along, even if you run into a situation where basically no one fixes the radio, would be a good thing to plan for... Just my 2 cents. --Jayron 32 13:24, 15 August 2023 (UTC)[reply]
Always good to keep in mind! I'm fully expecting them to fail on repairing the radio, the stakes are higher if they don't manage to call for help and need to find their own escape off the island. But if they beat the odds on the check (or somebody comes up with an ingenious alternate method for fixing the radio using a skill other than Mechanical Repair) I'm more than happy to let them succeed and introduce a new plot element through it. I just wanted to know how likely it was that happens or if I should implement some gimme like a Radio Operator's Manual they could find in a drawer. But at 1-in-3 odds I think I'm just gonna let them court lady luck. --Aabicus (talk) 19:05, 15 August 2023 (UTC)[reply]