The Effect of Rerolls on D6 Probabilities: A Visual Guide and Discussion

The Effect of Rerolls on D6 Probabilities: A Visual Guide and Discussion

Recently I was working on a sci-fi tabletop miniatures game, and while designing one of the factions decided to give some of their units a phasing ability. The lore and visual aesthetic of this was straightforward, but how to represent it in-game?

I greatly dislike +1 or -1 style modifiers, especially if there are a lot of them and they stack on top of each other. I could go on for an hour or so about why I think they are an overused lazy design that, among other things...

I digress. Maybe it's okay for things like cover but no modifiers like that for unit abilities! However I was open to the idea of rerolls. In order to make my game both fun and balanced, I needed to understand the statistical and probability effects these abilities and reroll rules would have on the D6 dice used in the game. Fortunately the mathematical calculations are not unduly complex or difficult, but I quickly found that staring at a handful of fractions and percentages on a text document didn't really help me "understand" what it would really do to an actual game with human players, not binary chalkboard spreadsheets.

So I made some graphs to visualize it. To help give context and comparison guidance. Since there wasn't anything of the sort readily available online already, I thought I'd share these graphs and a few of my insights regarding them. I like to think they are easily digested and clearly labeled, but I'll go through them to discuss. This whole rabbit-hole was started because of a miniatures wargame, but actualy the end results are probably just as or even more useful for regular board games. Games that use dice other than D6's aren't going to find these results very helpful though, so keep that in mind.

I will avoid going past five dice for the charts in this post since most target numbers and reroll effects quickly end up at or near 90% past five dice, with only things like a target number of 6 with a negative modifier not being 99.9% chance around ten dice or so. Relatedly, I didn't bother with 2+ and 3+ targets in these graphs because even with negative modifiers you are consistently likely to succeed even with only a single die and practically guaranteed to do so with a beneficial reroll or multiple dice.

First up are some simple charts of how likely you are to get at least one success with various reroll effects. The exact same information is shown in two different types of graphs. I think this is absolutely not redundant, because they give different impressions. 

Probably the first thing you'll notice is that the effect of the rerolls on probability changes drastically depending on what the target number is. In both directions actually, with the increasing or decreasing difference from a "plain roll" getting more and more exaggerated with additional dice. 

Maybe this is just un-asked for advice, but as a game designer I would highly recommend people try to avoid probabilities higher than 90% or lower than 10% if reasonable to do so. The purpose of this is simple: those likelihoods are not going to fail or succeed often enough to be fun or relevant. Unless your game involves rolling copious amounts of dice, then literally an entire game will go by start to finish without either player actually rolling two 1's in a row in a situation that needs it. It's only a 2.78% chance! That isn't zero or anything and if you roll a few hundred dice combined during the game sure it'll happen here and there. 

But that's the thing, it won't happen WHEN it's relevant. Coincidentally getting two 6's, like for example when you toss two dice, get a 2 and then a 6, reroll the 2 and get another 6. That was a statistically remarkable two 6's in a row! But you don't care, because for either die you actually only needed a 4. It's mildly notable from a mathematical standpoint but neither player will get excited about it or, Hell, even notice because they are focused on the end result and don't care if it was a 5 or a 6. And the few times that you absolutely need to make that final shot to take down the badly wounded monster, if you have a 93.4% chance or whatever then flubbing that roll won't feel like a tactical or strategic error or a risky gamble that didn't work out this time. It will just feel like an irritating and arbitrary fluke.

With that said, let's look at the relative effect of a reroll, rather than the overall effect on probability.

When I first started doing the math I was surprised by how little a difference rerolling 1's made. It's only 8.33% with Target 4+ and an even more meager 2.78% at 6+. That's just mild noise. That's not enough of a change to have any tangible or noticeable impact on a player's strategy or tactics or psychological warfare. Yeah, sure, if you are like Walmart looking at some quarterly revenue report than 8.33% or even 2.78% is such a difference that you can break out the champagne. But dice only land as a 4 or a 5, not as 4.515768 compared to 4.681293. And again, unless you are rolling entire cups full of dice and then spending all that time counting them and sorting them before you then spend another minute or two rerolling a bunch, it's not going to truly matter.

It's funny because forcing a person to reroll 6's, on the other hand, goes from a minor debuff at 4+ to a crippling shutdown at 6+! Even rolling 10 dice at a time doesn't give good chances.

So... is it better to do the "Reroll Hits" and "Reroll Misses" route instead? Yeah, personally I'd say so. Not only do they have much more noticeable effects on the game, but they are more mirrored regarding how much they alter the probability up or down compared to a single roll. Remember how I said neither player will care if a dice ends up as 5 or 6 if all they needed was a 4? That's the same mindset with rerolling Hits or Misses compared to a specific number. It makes it more of a Pass/Fail attitude than an arbitrary numerical emphasis.

I feel that, in regards to both fun and balance, the initially drastic effects of Hit/Miss rerolls on Target 6+ can be negated by overall good design that contextualizes it. Let me give a real, practical example:

You want to shoot someone. You start with a base 5+ target to hit them. But they are behind some cover, so that makes the target number 6+. Okay, you are fine with those odds, you still got a 16-ish percent chance per shot right? Ah, no actually... the target has a partial cloaking device that forces you to reroll hits when targeting them from a distance. Damn, now you only have a 2.78% chance of hitting them! That's basically impossible! This isn't fair! That cloaking device is overpowered!

Hold up now, it's not. You have other options. If you use a flamethrower or weaponized EMP or some other area of effect weapon, it will ignore the forced rerolls of the cloaking device. Or, if you haven't received any damage this round and choose not to move this turn, you can "Aim", with allows any unit to lower the target number by 1 when shooting. So it's back down to 5+, and you've got 11.11% chance. Or you could use a psychic power or hacking ability or something else that targets the mind, not the body, which would not only ignore the cover but also ignore the Reroll Hits effect for a pleasant 33.33%. So your assault troopers might have a hard time with that coaked fellow but your combat tech-priest is much better suited to taking him out. OR you could say that, well, I'll just shoot him with a heavy machine gun that fires lots and lots of bullets. That weapon has a built in Reroll Misses ability, so the two abilities cancel each other out and it's a plain 6+ target again. 16.67% or 11.11% or whatever aren't good odds but it's not as impossible as 2.78%!

Do you see how this works? It's not unfair or unrealistic or "gamey" that it's very, very difficult to shoot a person wearing a cloaking device who's also hiding behind cover from a distance. And even if you DO have a very hard time shooting them so what? The other player isn't going to win the game and achieve the Objective by hiding their units in the bushes near the edge of the map/board. If other aspects of the game overall or the specific scenario encourage movement or combined arms or "anti-camping" measures then that unit who is extremely difficult to shoot in that specific turn might end up a lot easier to hit later, or they WERE easier to hit a few turns ago and frankly it's your own fault for letting them get into such a great tactical position unopposed.

This absolutely goes in the other direction too. A unit ability that gives you Rerolls Misses which might at first seem like an unfair and unfun "he always hits me why does he even need to roll for fucks sake..." could be balanced and managed if you did better about taking cover, arranging rock-paper-scissors match-ups in your favor, focused on the scenario objective instead of bloodthirsty combat, made your army list more well-rounded defensively, etc etc.

Let's look at the impact of rerolls from a different angle. Previous charts were focused on the likelihood that you'd score at least one success. But obviously that is not the only thing that matters. If you are hacking away at a monster with 5 HP, or need a cumulative amount of successes to complete an arcane ritual regardless of how few or many turns that takes you, then you care about the average number of the successes more than you care about the likelihood of getting a success. Let's see the charts!

Now naturally you can't get 0.5 or 1.82 successes in real life, since you can only get whole numbers with D6 dice. But these give some telling insights none-the-less. The biggest difference between these charts and the earlier ones is that average number of successes is linear, not binomial like the % chances. Rolling three times against a 5+ target does NOT make you three times as likely to succeed. Sadly for our puny human brains probability doesn't wok that way. But the cumulative results would actually be three times as high on plain addition. Again we see that the Reroll 1's effect has very little impact, especially at a single die but even all the way up to five. The Reroll Hits and Misses ones have more of a noticeable impact.

But I think this shows that, if your game or your special ability or whatever does involve tossing substantial amounts of dice, the effects of a beneficial rerolls ability start giving you a hefty amount of successes. With ten dice against 4+, with rerolling misses, you could casually assume you'll get five or six hits. Certainly you'll get at least two or three, right? Conversely, even if you start avalanching all the D6 dice you have nearby at the same time you'll never have good odds of getting more than one success against 6+ with either negative reroll type. But you stand a reliably good chance of getting at least one if you can reroll misses!

But what's that you say? You don't care about "successes" you care about the total number of pips from a roll? Well, I mean, that's not a very common thing in most games, but yeah sure it's valid. Dungeons and Dragons famously uses a few D6s added up for the starting ability scores of characters, though that game also uses D20's not just D6's so... Yahtzee? That game is a whole 'nother can of worms regarding probability...

Maybe people don't use it very often because it's not common knowledge how rerolls influence that! Good thing I'm here! Though actually it's just because it takes a lot more time and effort to do all that addition over and over again, especially if you have lots of dice. And then there is usually a bunch of record-keeping associated with it... Still, it's just more basic yet tedious math that I figured out before writing this so that you don't have to! 

Now, when looking for as high a total as possible rather than a specific number or better, you don't have any "Hits" or "Misses" but you can calculate the odds if you could voluntarily reroll "low" numbers or be forced to reroll "high" numbers. Which you would, logically, define as 1/2/3 and 4/5/6 because if you rolled a 4 initially you are more likely to roll something worse or the same than you are to reroll that into a 5 or 6.

So.....

As you can probably see, the effect is very minor unless you are rolling lots of dice and adding the pips up. Which quickly becomes cumbersome and boring after five dice and unless there is a substantial difference between a result of 17 and 19 in your game, it probably won't make a meaningful impact regardless of what beneficial or hindering rerolls you have. Having visual evidence of that is helpful though, and again perhaps your game actually does have tangible consequences between a total result of, say, lower than 10 or higher than 10. If so, and if in your game you routinely roll three dice, against that threshold rerolling highs and lows would actually come into play in a meaningful way frequently. But it might be worth considering just adding or subtracting a die instead, to cut down on time and brain power spent on rerolls.

I can hear the people in the back... "Yeah, that's swell dude, but in MY game we don't use target numbers OR cumulative totals. We use opposed rolls, where both players roll one or more dice at the same time."

Alright listen here you lil' shits... 

I mean, uh, that is, mathematically speaking that would look similar to but not identical to the graphs and discussion we already went through, just with the "target" numbers decided at the last moment by whichever die happened to stop moving first. I understand it would be a bit more complicated than that regarding % probability, since unlike the stuff we've gone over so far the temporary "target" results one players needs to beat the other player might be affected by different reroll rules, which would alter the overall impact of "your" reroll effect on the probability you'd win in the end. 

That is, while the above graphs could give you accurate info on your chance of victory or defeat after one of the opposing dice stops moving, that isn't the same as knowing your overall likelihood of winning before taking the tactical or strategic action that triggers the opposed roll. 

I could do the math, but it would be difficult to display that information without cumbersomely large or dense graphs, or a large number of simpler but very specific use-case graphs. There are too many variables. What if different players roll different numbers of dice? Is there only one "round" of rerolls after the initial toss or is it more like each player is allowed a reroll independently? Like, we both have a Reroll Misses ability. I roll a 4 and he rolls a 3. So he gets to use his reroll, and then gets a 6. Do... I get to reroll now too? No, it's only a second chance if I initially lose the toss? What if we both roll the same number in the first toss? That is a whole lot of very specific calculations (some of which are tricky to reduce into a simple "X% chance I win" numerical value) to display in a single post.  

In many ways the numbers would be very similar to what is above anyways. So if you intend to have opposed rolls in your game, just look at the graphs here and imagine a few minor changes up or down. And maybe reconsider the wisdom of opposed rolls with rerolls, because that is going to be a nightmare to even playtest, let alone reliably balance.

So... anyhows,

What have we learned? Well, nothing really since I did all the math for you and didn't explain how I did it or how I implement these rerolls into my game/s. But personally I feel a lot more informed and confident in my decision to emphasize rerolls over generic +1 or -1 modifiers. I also feel comfortable recommending that the Reroll Hits and Reroll Misses way is better than the Reroll 1's and 6's idea. Mostly because it is more symetrical in the positive or negative impact and also less of a trivial difference, but also because I feel that the Pass/Fail mentality is superior to the number specific mindset.

Dolling out these reroll abilities too liberally or arbitrarily would bog down the game and be very difficult to balance. If four-out-of-five units in your army and three-out-of-five in your opponent's army are rerolling on a regular basis, we are going to be spending literally twice as much real life time or worse on a very basic aspect of gameplay. And it can feel very arbitrary or immersion breaking to see such an effect on the dice that doesn't have a plausible or rational justification in the lore or aesthetics. 

My Lunar Elite veterans of the Marson IV campaign get Reroll Misses on their Fear rolls or Initiative rolls or whatever. Okay sure, that makes sense, they are experienced combatants so might be a little quicker on the rollout or need less time to react to things than a fresh recruit because unlike the new guy they've already seen horrible cosmic abominations before and defeated them. But that guy has a jetpack... so... why does he make your opponent Reroll Hits in melee combat again? 

Thus, in summary, jetpacks are awesome, +1 modifiers are to be avoided when possible, and I think I got all the math correctly computed and accurately placed on my graphs. If you actually care about the formulas or want to double check my findings, I am totally happy to write them out in a comment down below! And yeah, I'd love to hear your thoughts on rerolls, or know if anybody but myself actually benefits from these data visualizations. Keep making and playing games!

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Bonus Round!

Here are three example formulas. For these examples I'll use the Reroll 1's effect. The process is mostly the same for different target numbers and reroll effects. You figure out the formula for a single die, then use binomial calculations or arithmetic to work out subsequent additional dice.

If, say, the target number is 5+, and you reroll 1s, the formula would look like this, at least before any condensing or simplification:

((2/6)+((1/6)*(2/6))

Not so hard. It results in 0.38̅ or ~39% chance of at least one success. Now, since each die of a set acts individually, that is, you may roll them at the same time but the results of any specific die do not interact with or change the results of other dice, a straightforward Binomial Cumulative Distribution Function gives you a percent chance for N number of dice.

So if, say, we wanted to see what our chances are with the Reroll Ones ability and a target number of 5+ when we roll four dice... it would look like....

Y=F(1∣4,0.38̅)=1∑i=0(4i)0.38̅i(1−0.38̅)(4−i)I(0,1,...,4)(i) 

*which may or may not display correctly on your screen*

So... ~86.05%

Math!

Now, for the average total, on a normal D6 die you have equal chances of getting any result so the average total is ((1+2+3+4+5+6)/6) or exactly "3.5". But if you are forced to reroll 6's or allowed to reroll 1's, then you actually have a slightly more than 1/6 chance of getting the other numbers and only a 1/32 chance of getting the 6 or 1 a second time in a row. So the average total is different. Same idea, with a few more parentheses, for rerolling Highs and Lows.

If we sought the average cumulative total for rolling 3 dice if we are Rerolling 1's, the formula would look like this:

(1*((1/6)/6))+(2*((1/6)+((1/6)/6)))+(3*((1/6)+((1/6)/6)))+(4*((1/6)+((1/6)/6)))+(5*((1/6)+((1/6)/6)))+(6*((1/6)+((1/6)/6)))

Simplified and turned into a decimal number, it is 3.916666666666666. No binomial crap here, more dice just stack up more pips, so that number times three, or...

11.75 as the average total in that situation. 

Ta'da!