Which is better? +1 to Hit or extra hit on 6

[ValourousHeart]

Pretty straight forward question.

Is it better to have +1 to the hit roll or 1 additional hit on an unmodified 6?

Does that change if the weapon has a random number of shots vs a set number of shots?

Does it change when considering weapons with 1 shot instead of weapons with multiple shots?

[Rogue]

For any individual dice, it makes no difference - +1 to hit and extra hit on a six are functionally the same effect.

[imagine rolling six dice, and getting 1, 2, 3, 4, 5, and 6. If you usually hit on 3s, that's four hits. With +1 to hit, the 2 also hits and you score five hits; or, with the 6 generating an extra hit, you still get five hits. The chances of rolling a 2 or a 6 are the same, so the outcome balances.]

 

That said, there is a difference between them. +1 to hit converts potential misses into hits. Whereas exploding 6s converts hits into more hits. One mitigates your failures, the other magnifies your successes.

 

 

Imagine a model with three attacks, hitting on 3s. The simple maths says you'd expect 2 hits, but the spread is more useful here:

• 4% chance of no hits
• 22% of 1 hit
• 44% of  2 hits
• 30% of 3 hits.

Two hits is still the most likely outcome, but we can see the slightly better chance of scoring three hits rather than one.

 

 

If we now hit on 2s instead, you get this:

• 0% chance of no hits (well, 0.4%)
• 7% of 1 hit
• 35% of 2 hits
• 58% of 3 hits

More accurate, especially at the top end. There's virtually a 100% chance of landing at least one hit, and a 93% chance of at least 2 hits (against just 74% when hitting on 3s). The most likely outcome is 3 hits.

 

 

And if we go back to hitting on 3s, but with 6s generating an extra hit, we could potentially score up to 6 hits in total (on a triple 6):

• 4% chance of no hits
• 17% of 1 hit
• 31% of 2 hits
• 29% of 3 hits
• 15% of 4 hits
• 4% of 5 hits
• 0% of 6 hits (again, 0.4%)

This time, we're back to 2 hits being the most likely outcome, with 0 hits and 5 hits being equally likely, and 6 hits being a remote possibility. This is the gambler's option - less reliable, but a potentially greater reward if it comes off.

 

 

Cumulatively, it looks like this (exploding 6s on the left)

 

6 hits     0.4%     0%

5+ hits   4%        0%

4+ hits   19%      0%

3+ hits   48%      58%

2+ hits   79%      83%

1+ hits   96%      99.6%

 

Which means you have a better chance of scoring 1, 2 or 3 hits using the +1, but a 19% chance of outperforming anything the +1 can achieve if you use exploding 6s instead.

 

 

So I guess the real question here is: "Do you feel lucky?" If you prefer cold, logical efficiency, take the +1 to hit. But if you feel that the dice are hot (or you play orks), exploding 6s could be for you.

Edited June 13, 2021 by Rogue

[ValourousHeart]

Thank you this was very helpful.

 

I think for any 1 and 1d3 shot weapons I'm going to play it safe and go for the +1 to hit.  I would like to feel lucky, but my dice fail me at the most inopportune times.

 

But I think for the weapons with a more reliable number of shots I'm going to risk it and go for the exploding 6s.

[Gorgoff]

I may add something to the already very elaborate answer of Rogue, which we keep in mind during my answer.

 

It depends on two things in my book:

 

1. How many shots do your gun have and

2. What is the to hit number?

 

For a one shot weapon I'd always take the +1 to hit because every shot counts but the more shots you have the more appealing exploding 6es get.

At a certain point it makes it more reliable to take the exploding 6es in order to assume what is going to happen.

For instance if I have 36 shots I would assume that I get 12-ish hits with the exploding sixes etc.

 

The second topic is how good you hit already. I think it is more useful if you looking at 3+ to 5+. 

[Rogue]

The more shots you have, the less it matters which bonus you have. As you roll more and more dice, they'll tend towards an even distribution (with tend being the key word here - aberrant results are still going to happen); and as the distribution evens out, you're likely to gain as many hits from the +1 as the exploding 6s. 

 

At the other end of the scale, a single-shot weapon can either go for reliability, or gamble on potentially landing two hits and doubling its effectiveness. Both options have a one-in-six chance of benefitting from the respective bonuses, so it's a preference call.

 

Regarding the point about a model's base chance to hit, that's interesting. I ran the maths on 6s to hit (as the other end of the scale to 3s to hit). Again, I've gone with 3 attacks, because it makes the maths a bit easier.

 

3 attacks, hitting on 6s (base-line numbers)

• 58% chance of no hits
• 35% of 1 hit
• 7% of 2 hits
• 0.4% of 3 hits

With just three attacks, the most likely outcome is missing them all, and there's only a 7% chance of getting 2 or more hits.

 

With a +1 to hit bonus, we now hit on 5s, which looks like this:

• 30% chance of no hits
• 44% of 1 hit
• 22% of 2 hits
• 4% of 3 hits

This pretty much halves the chance of missing completely, and makes it eight times more likely to get all three hits (1/216 vs 8/216, which is lost slightly in percentages due to rounding).

 

On the other hand, back to hitting on 6s that then explode into two hits looks like this:

• 58% chance of no hits
• 0% of 1 hit
• 35% of 2 hits
• 0% of 3 hits
• 7% of 4 hits
• 0% of 5 hits
• 0.4% of 6 hits

Because we're hitting on 6s anyway, any hit becomes two hits, so we can't get 1, 3 or 5 hits. Otherwise, the chances are the same as the base-line numbers, just with twice as many hits attached.

 

Here's the cumulative table again.

 

6 hits     0.4%     0%

5+ hits   0.4%     0%

4+ hits   7%        0%

3+ hits   7%        4%

2+ hits   42%      26%

1+ hits   42%      70%

 

This is a different balance to the parallel version hitting on 3s. There, having +1 to hit gave a better chance of scoring anything up to three hits. Here, +1 to hit gives you a better chance of getting a single hit, but exploding 6s gives you better odds all the way up after that.

 

So my tentative suggestion is that exploding 6s becomes increasingly useful as your basic chance to hit decreases (once again, good news for orks). 

 

 

For the visual thinkers amongst us, I've knocked this up:

 

 

Clearly, you'll do better if you start with a better WS or BS. Equally evident is that either bonus is better than neither (the red lines). The interesting bit for us is the blue and green lines - we can see that they cross in different places. It suggests that if you have a good basic WS or BS, then +1 to hit is better for longer. But for a low initial WS or BS, gaining whole extra hits is quickly better than your chances of rolling them on the dice. 

Edited July 7, 2021 by Rogue