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In a scenario of 5 rounds in level 1, where spell range doesn't matter, and the enemy's AC gives you a hit chance of 50%. Which is better: casting Witch Bolt, or spamming Fire Bolt for 5 consecutive rounds?

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  • \$\begingroup\$ Sorry, it's in level 1. \$\endgroup\$ – Maxoy Feb 8 at 21:52
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    \$\begingroup\$ related: How effective is Witch Bolt? \$\endgroup\$ – G. Moylan Feb 8 at 21:53
  • \$\begingroup\$ I implore you (and anyone else who reads this question) to read the answers to the question linked above. The maths alone does not tell the full story here. \$\endgroup\$ – Miniman Feb 9 at 1:46
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At Low ACs, Witch Bolt is better. At high ACs, Firebolt is better

The trick with Witch Bolt is that the initial casting of the spell isn't spammable at level 1. You only get to use it twice, and if either casting fails, the damage fails completely. Conversely, with Firebolt, you can keep using the spell even if prior uses failed. So if Witch Bolt successfully lands, you have a source of guaranteed damage, but if it fails, you can't just endlessly recast it. You can try again the next turn, and... well.

The average damage from these spells, over five rounds, looks like this:

\begin{array}{|l|r|r|r|r|r|r|} \hline \text{Name} & \text{To-Hit 1} & \text{To-Hit 11} & \text{To-Hit 14} & \text{To-Hit 16} & \text{To-Hit 18} & \text{To-Hit 20}\\ \hline \text{} & \textit{(AC 6)} & \textit{(AC 16)} & \textit{(AC 19)} & \textit{(AC 21)} & \textit{(AC 23)} & \textit{(AC 25)}\\ \hline \text{Firebolt} & 27.500 & 15.125 & 11.000 & 8.250 & 5.500 & 2.750\\ \hline \text{Witch Bolt} & 31.200 & 16.575 & 11.700 & 8.450 & 5.200 & 1.950\\ \hline \end{array}

The To-hit 11 is your case: you'll hit on an 11 or better, meaning you hit 50% of the time. AC values assume +3 to primary spellcasting modifier.

The other consideration is that the damage-over-time effect of Witch Bolt cannot crit, but the repeated uses of Firebolt do each have a chance to crit. Outside of very high AC creatures though (To-Hit of 18 or 20, which correspond to ACs of 23-25; very unlikely for a Level 1 character) Witch Bolt will end up being better.

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  • \$\begingroup\$ I've downvoted you - your maths is fine, but you're unconditionally recommending Witch Bolt, and that's bad advice. \$\endgroup\$ – Miniman Feb 9 at 1:43
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    \$\begingroup\$ @Miniman Uuuuhm, two things. First, I don't "unconditionally" recommend Witch Bolt: I'm very clear that in some circumstances (high AC creatures), Witch Bolt isn't actually better. Secondly, comments are for workshopping/improving answers, not for arguing about their merits. If you think you have a better answer, you should be posting it yourself. \$\endgroup\$ – Xirema Feb 9 at 2:23
  • \$\begingroup\$ I don't have a better answer, and I'm suggesting that a disclaimer about how mathematical analysis ignores most of the drawbacks of Witch Bolt would improve this one. \$\endgroup\$ – Miniman Feb 9 at 2:26
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In this scenario, witch bolt is better

Fire bolt:

In 5 rounds, hitting half the time, this will deal 5d10 / 2 damage, so an average of 13.75 damage.

Witch bolt: (assuming you have 2 spell slots to spare)

There is a 50% chance of dealing 5d12 damage, if you hit the first attack, and a 25% chance of hitting with your second attack and dealing 4d12 damage. Then, there is a 25% chance you miss with both those attacks, so 0 damage, as you do not have spell slots left.

Given the equation (5d12 + ((4d12 + 0)/2)) /2,

The average will be 3.5 d12, or 7d12 divided by 2, it gives us an average of 22.75.

In Conclusion:

Witch bolt deals more damage on average, and even after you use all your spell slots, you can use your action to do other things.

However, fire bolt is a cantrip, and though witch bolt deals more damage on average, it doesn't expend those valuable resources. Another consideration is the numerous ways that witch bolt can be stopped, but I did not include this due to the nature of the question.

Note:

I did not include critical hits, as this just complicates the math and is negligible. It would, however, increase the average damage output of fire bolt more than witch bolt.

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That depends on your level.

At low levels:

It's obvious that fire bolt is strictly inferior at 1st level; not only do you have to roll every attack, even if you hit every time, your expected damage is just lower. (Witch bolt's all-or-nothing nature doesn't change its average damage output; it just makes it extremely swingy.)

But your fire bolt damage goes up with level, and that's where it gets interesting.

At mid-levels:

Let us assume you hit with the witch bolt. In that case, you deal a total of about 33 damage across 5 turns. Your fire bolts have half a chance to hit and, at 5th level, deal 11 damage on a hit, yielding an average total damage of 28, which is still below the witch bolt, but only just. At 11th level, with 3d10 per fire bolt, your expected five-round damage comes up to 41, which makes it the better deal.

But why did I assume witch bolt hit in the first place? Put simply, opportunity cost. Witch Bolt has a single action opportunity cost -- if you hit with it, you're guaranteed to keep dealing damage with your actions. If you miss, you don't lose out on all that damage; you can spend the next four rounds doing something else instead.

But wait, there's less!

What if we assume that, if you miss the witch bolt attack, you'll switch over to using fire bolt for the remaining four turns? In that case, your total expected damage at 5th level would be 27.25, while your expected damage from just spamming fire bolt is 27.5, a difference small enough to be not really worth discussing. But then witch bolt also has the opportunity cost of a spell slot, so at that point you really might as well just spam cantrips.

So basically, if you play smart, witch bolt is better only until your cantrips upgrade at level 5, and thereafter, it's not worth the effort unless you upcast it using a stronger spell slot.

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Which is better depends on the target

For example, against creatures with 1hp, Fire Bolt is better because any hit is a kill and cantrips don’t use spell slots.

Against a single creature that will still be alive after 4 rounds, Witch Bolt always does more damage. However, against such a high HP creature you would not be using either of these spells — you’ll be using your 4th and 5th level spells because this is a CR9+ creature.

Witch Bolt does more damage overall since the chance to hit is the same and the damage is more. That you no longer have to roll to hit again once you hit with Witch Bolt is just a bonus. Then, if you run out of spell slots you just switch to Fire Bolt.

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Witch Bolt (unless elements matter)

Given range doesn't matter, both spells use a ranged attack roll. On a hit Fire Bolt does 1d10, Witch Bolt does 1d12 + you are guaranteed to hit the next few rounds if your concentration is unbroken. Even if it is, you're still doing 1d12 instead of 1d10.

If the element of the spell matters, then use whichever element is better.

  • If Lightning is resisted, Witch bolt drops from 6.5 to 3.
  • If target is vulnerable to fire, Fire Bolt jumps from 5.5 to 11.

Note that ignoring range (and things like cover) ignores both the drawbacks of Witch Bolt (low range, you need continual line of sight) and the benefits of Fire Bolt (long range), so it isn't a great comparison for the general case.

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    \$\begingroup\$ Also note that with Witch Bolt, there's a flat 50% chance to hit, and then that 1st level spell has been expended. With the Fire Bolt cantrip, you have five 50% chances to hit. \$\endgroup\$ – Davo Feb 8 at 22:17
  • \$\begingroup\$ @Davo in the case that you miss with all your Witch Bolts, you can fall back on Fire Bolt spam anyway. But it is irrelevant, as Witch Bolt still ends up better. \$\endgroup\$ – GreySage Feb 8 at 22:23

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