Chances of getting three successes

I was listening to the Lady Blackbird podcast and I became curious to what where the chances of getting three successes with a number of dice. I remember figuring out two successes for Burning Wheel but since Lady Blackbird (at least in the podcast) uses three, I went back to the drawing board.

3: 12.5% (this one's obvious: half * half * half = 12.5%)
4: 31.25%
5: 50.0%
6: 65.625%
7: 77.34375%
8: 85.546875%
9: 91.015625%
10: 94.53125%
11: 96.728515625%

My favorite chance of success as a game master or as a game designer is about 2/3. That means having six dice in Lady Blackbird. That didn't seem that common in the podcast, but I don't know.

I also came up with a pretty neat algorithm to finding it quickly:

(define (probs dice)
(* (/ (let loop ((dice dice) (obst 3))
((zero? obst) (expt 2 dice))
((zero? dice) 0)
(else (+ (loop (sub1 dice) (sub1 obst))
(loop (sub1 dice) obst)))))
(expt 2 dice))

In other words, the program creates a binary tree of all die results, counts all branches that had at least three successes, and divides that number with the total number of branches, which is always 2 ^ the number of dice. E.g. with four dice there are going to be sixteen branches (2 ^ 4 = 16), five of those have three or more successes, and five divided by sixteen is 31.25.

The cool thing is that as soon as one branch has had three successes (rephrased as reduced three obstacles to zero), I know that all of its sub-branches are going to be successful as well so I can just sum it up without going through them.
That said, there are still a lot of low-hanging fruit optimizations available, like I probably call (loop 4 3) a bunch of times. But sometimes memoization can be more expensive if the numbers are small enough.


  • edited July 2015
    Anydice is good for this.

    N successes on K dice would be "output Kd{0,1}" and set the view mode to "at least".

    Here's an example for 4 dice (you have to change the view mode), showing 31.25% chance of at least 3 successes.
  • Oh, this is good. Very good.

    (Sandra, if you're interested, you should see my Lady Blackbird Otherkind hack. It's another fun way to use the whole Blackbird thing.)
  • Is it this?
    I don't get it... maybe I should read Otherkind :D
    I'll look at it more closely, Paul. First impression is that Lady Blackbird as written already has a very high failure rate when rolling.
  • Naw, re-reading it it's pretty self-explanatory. Looks fun.
  • edited July 2015
    It's a LOT of fun. I've played it several times now, and it's a blast. (Like I say in the thread, the only challenge is the handling time, but everyone I played with felt it was worth it - and most of the handling time is in Blackbird already, anyway: it's choosing Tags that tends to slow the game down most, in my experience.)

    As for chances of failure, it's a bit different. You have much BETTER chances of success (in fact, you can almost never fail, if you make it your priority) than in regular LB. The difference is that your successes are complicated and messy: you get Lost and Injured while situations grow more and more dangerous. Much like rolling 7-9 all the time in a game like Apocalypse World.

    I wrote a whole hack of Blackbird using that system (including a different starting situation)... I should probably edit it and share it with everyone here sometime.
  • Let's say things have one danger usually.
    To get two sixes, that's 7% on three dice. And only 20% even on five dice!
    Even with seven dice it's only 1/3 chance. Paul, this is unreasonable?!
    (I get that you can also flip to ones but then you get a condition.)
  • Ah, but you only need one success!
  • For instance, if you're in a fight with Uriah Flint, and you want to kill him... you just need one 6 or one 1 in order to do so.

    The other dice just determine whether there's a complication which also happens.
  • The Danger is that Naomi flies of into space! That's not exactly missable :D
    What's a typical number of dice in your experience with the hack, Paul?
  • edited July 2015
    Well, the Dangers vary from task to task, right? Sometimes you won't care about them; sometimes they'll be really important.

    It's the GM's job (or the group's; it's often good to brainstorm Dangers together as a group) to balance the severity of the Dangers with the scope of the Goal.

    The typical number of dice is the same as in Blackbird (although in my hack I reduced it a tiny bit by reducing the number of available Tags, and that felt faster and better). I seem to remember people rolling 6-9 dice most of the time.

    Maybe it would help to think of it this way:

    1 Success - partial success (a 7-9 in AW terms)
    2 Successes - full success (10+)
    3-4 Successes - critical success

    That's more or less what it feels like.

    In the Naomi example, she flies off the back of the ship, which means we need a follow-up challenge to rescue her (does the crew turn back to rescue her, or keep flying?), or maybe she gets captured by the Imperials and now needs to try to escape herself, or some other such thing.

    And keep in mind that you can often check off a Condition in order to upgrade your success dramatically. (Almost half the time.)
  • But a 10+ with a zero stat in PbtA is still more likely than two successes on four dice; it's slightly less than two successes on five dice.
    So if people are rolling 6-9 dice in Otherbird, that's a little better, but you're still more likely to fail than to succeed. (With 9 dice, 45 chance of two successes.)
  • edited July 2015
    Well, sure. I designed it that way.

    However, keep in mind that you also have your Pool dice available, your friends can lend helping dice, and you have to factor in the odds of taking a Condition and using your 1's.

    After playing it, we found we preferred it with fewer dice, but it works fine "as is" (with standard LB characters). (I find that 45% chance, for instance, overly high - it's like an AW character rolling at +2.)

    Being able to name your Goal means you have more agency as a player and as a character than you would in most other games. You want something, you go for it and you will almost certainly get it. It just won't be a clean victory!

    I think you're really looking at this the wrong way 'round if you think a single success is a "failure". It's quite the contrary. (If you'd like, we can "play out" a hypothetical situation, and you'll see what I mean.)
  • Yes, please let's do that Paul!
    Let's say you're trying to wrest control of The Scolded Apothecary from Captain Ginny and the danger is that Naomi will get shot in the battle. Would that be an appropriate example?
    What do you do?
  • edited July 2015
    Haha! Excellent. What is the Scolded Apothecary? A ship? A book? An inflatable travel companion?

    Just for fun, let's say I'm Snargle, and I have 2d6 Pool points.

    (Rolled: 2d6. Rolls: 5, 3. Total: 8)

    Ok. Now, for starters, if Naomi was a PC, we probably wouldn't name this as a Danger. (Or, if we did, we'd check with the player if s/he was OK with it, and then probably Naomi would throw in some helping dice.)

    We have a skirmish of some kind, and I decide that I want to grab the Apothecary, whatever it is - that's what I want in this scene.

    It sounds like there's some wrestling and scratching and biting going. So maybe I'm rolling Goblin, with applicable Tags being Warp Shape, Agile, Quick, Tumbler, and Teeth & Claws.

    So, one die for starters (1), plus one for the Trait (1), plus one per Tag (5). So I'm starting with 7 dice. Maybe I'll throw in 3 from my Pool, for a total of 10.

    Now I'll roll them.

    (It turns out you can't edit a post with a die roll without the "cheating" message showing up. Sorry about that! Probably a smart feature.)

    POTENTIAL CHEATING DETECTED. If it appears that dice were rolled in this post, they may need to be disregarded as fabricated.
  • Rolling the dice:

    (Rolled: 10d6. Rolls: 2, 2, 6, 2, 2, 6, 3, 3, 1, 3. Total: 30)
  • Ok. Snargle for the win!

    Let's try a few more rolls, just to see some typical results.

    1. (Rolled: 10d6. Rolls: 6, 5, 6, 5, 1, 1, 2, 1, 2, 3. Total: 32)

    2. (Rolled: 10d6. Rolls: 4, 2, 6, 5, 1, 2, 6, 1, 2, 1. Total: 30)

    3. (Rolled: 10d6. Rolls: 3, 5, 5, 4, 5, 6, 1, 6, 6, 3. Total: 44)

    4. (Rolled: 10d6. Rolls: 1, 6, 2, 1, 4, 2, 2, 5, 3, 3. Total: 29)
  • edited July 2015
    Ok, so far all successes (perhaps beyond statistical expectations, but Snargle's pretty good at this kind of thing). But let's look at the last one, that one's interesting. I have one 6 and two 1's.

    My choices:

    1. I manage to get the Apothecary. Naomi gets shot in the process.
    2. I give up on the Apothecary during the battle, in order to protect Naomi. (Snargle's hoping for a kiss afterwards, perhaps? Not likely, something tells me...)
    3. I get what I want AND manage to keep someone from shooting Naomi by pushing her out of the room, but I get Trapped on the bridge as the airlock doors shut (a Condition).

    That's pretty much how it goes.

    I don't know about you, but I want way, way less of the first type of roll and more rolls like that last one. That's some exciting stuff; and I can pick and choose moments when I want use a lot of my Pool dice to (hopefully) succeed when I really need to.

    The latter type of outcome feels much more like a fun Steampunk adventure than a straight success (to me, anyway). The heroes succeeded... but now here's a new problem! Naomi is bleeding out, we need to get to the planet's surface, where there is a healing pod, before she croaks! Snargle took control of the Apothecary... except now he's trapped on the bridge, and the ship's on fire. And so on.
  • edited July 2015
    Another thing:

    If you have mixed success in your game (like AW or this hack), you can't have 66% success rates for full success. That leaves almost no room for failure or the outcomes you're hoping to have.

    In that kind of game, full success is more like a critical hit - a moment when you cheer because, wow, that was cool, well done, Snargle, you really pulled it off! (Think about your favourite movie and you'll notice there are very few such outcomes - they're rare and far between.)

    It's also important, however, to pick the right Dangers. A major character getting shot (especially if it's in-genre for them to possible die from it) isn't a typical Danger, it's a really unusual, high-stakes one. It's the kind of thing that happens at the climax of an episode. So that means players will probably be using more Pool dice, as well - and the danger of something really bad happening is very, very appropriate.

    (In fact, when I play this game, I often use two Dangers - and sometimes three - in high-stakes situations, I get rid of the Basic die - you only get Pool dice and Trait/Tag dice - and remove a few Tags from each Trait.)
  • Ah yeah, the notion that you wouldn't typically put Naomi in such extreme danger if she was a PC was my first mistake. (And severely underestimating the amount of dice -- though this came from the first couple of rolls in the podcast -- was my second mistake.)
    I kinda figured The Scolded Apothecary as an aether ship of the typical LB sort :D
    a free roamer kind of competing with The Owl.

    Thanks for hashing this one out with me. All is well now that I've seen my errors :D
  • (Oh, it's fine to put Naomi in danger - but only if she's doing something really dangerous AND it's her roll, not Snargle's. Perhaps you were reading my example in that thread as suggesting that Naomi's player was trying to escape on the Owl, and the Danger was that she would blown out into space? That's a pretty intense Danger, and it arguably disqualifies the Goal of the action. That's a no-no; it should always be possible to achieve the Goal even when the Dangers can't be avoided. That's up to the group, though, to remember: there's a bit of a learning curve to naming good Dangers, though it doesn't take very long.)

    (And the Scolded Apothecary is an excellent name for a ship.)
  • Perhaps you were reading my example in that thread as suggesting that Naomi's player was trying to escape on the Owl, and the Danger was that she would blown out into space?
    Worse: that she was a PC, but that her life was on the line on another player's roll!

  • In that kind of game, full success is more like a critical hit - a moment when you cheer because, wow, that was cool, well done, Snargle, you really pulled it off! (Think about your favourite movie and you'll notice there are very few such outcomes - they're rare and far between.)
    I'm weird when it comes to movies because I really like seeing competence and success on screen. :)
    Heist movies where everything goes right are cooler to me than where they "fail forward".
    I love success but real success needs a risk of failure to matter.

  • That's actually an interesting design point, I think. A game where success is the usual outcome, but the risk of failure is nevertheless very real... is a rare thing, if I've ever seen it at all. (Video games come to mind, of course.)

    That might be a fun design challenge.

    Having said that, if that's your groove, I would advise you not to play any games based on Otherkind dice, like Ghost/Echo and all the way through to Apocalypse World. (The whole "fail forward" paradigm and "success at a cost" dominates modern RPG design, at least on the 'indie' side of things.)

    Are there any games which you feel do this really well? (I can imagine that some versions of D&D and related games might do this well: competent characters who mostly succeed, but with great danger at stake in case of a failure.)

  • D&D comes to mind yes. 5e is my favorite game and the binary success/fail is a big part of that (because we switched from Fate where the four outcomes really took center stage, and I was itching to go back to a plain success/fail system). A lot of the games I groove on have binary success/fail, even hippie games like DramaSystem or Fiasco to some extent.
  • "Push your luck" mechanics like in Black Seven often lean towards success in each individual roll, but make you roll many many times the further you pursue your goals, so that eventually by the operation of statistics you can expect at some point to fail.
  • Is that just a question of rolling many many times, always with a high chance of success? Or is there something more involved at play?
  • Sandra,

    If you like those "successful heist movies", why do you say that your preferred success rate is around 66%? What happens if it's higher?
  • Is that just a question of rolling many many times, always with a high chance of success? Or is there something more involved at play?
    Correct. Like, it's a heist that requires many steps and feints and stealthy actions and knocking people out with blackjacks, and you're SO GOOD at everything, but there's SO MUCH to do in order to achieve all your goals/grab all the loot/etc. So you decide if you're going to "push your luck" in order to get more of what you want, versus taking the shortest/safest approach.

  • I'd say both Riddle of Steel and 3:16 have reliably produced fairly high overall success, real sense of danger games for me.
  • I'm cool with higher :D
  • Matt, want to explain how/why? (For those of us who haven't played those games.)
  • In 3:16, you can narrate a flashback to save yourself. But you only have a limited amount
  • The combat system is brutal in TRoS and it's not hard to die, per se, but if you are careful and fight for what you believe in you are pretty kick-ass.

    3:16 gives you various ways to save yourself, such as the aforementioned flashbacks (also later on certain desperation moves) but eventually they run out or you don't have one available.
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