# Is it better to dash or dodge across an open field?

Suppose I have to run 300 feet across an open field from one area of full cover to another. The entire time I'm in the open, I will be taking incoming ranged attacks, and I want to minimize the number of attacks that hit me as I cross the field. My speed is 30. I have 2 options:

1. Take the dodge action each round and move 30 feet per round for 10 rounds, thus facing 10 rounds of attacks at disadvantage.
2. Take the dash action and move 60 feet each round for 5 rounds, thus facing 5 rounds of attacks without disadvantage.

Which of these 2 options results in fewer attacks hitting me on average? If the answer depends on the roll needed to hit, when is it better to use one or the other? Alternatively, is there a more complex solution using only "basic" actions (i.e. only actions available to an unarmed commoner) that does better than either of the above?

For simplicity, you can assume there is 1 incoming attack per round, although the result should be the same for any number of attacks. You can also assume that all attacks have the same modifier. You can also assume that the attacks are always made within the short range of the weapon, so there is no disadvantage from long range. (For this last point, imagine the 300-foot run is parallel to a wall lined with enough archers to cover the entire path in their short range, or more simply, just imagine a single longbow archer with the sharpshooter feat.) I am not overly concerned with the effect of critical hits, but if you want to take them into account, you can assume that a critical hit deals double the damage of a regular hit (even though this is not true in general).

### There's a third option - and it's potentially much better

The other answers have dealt capably with your 1 vs. 2 scenario already. To recap what's been established elsewhere:

1. Dodge every round - every attack against you has disadvantage
2. Dash every round - suffer half as many attacks against you

Which is better? It depends on the enemy's likelihood 'to hit'.

But there's another way, a third option that's normally superior to both of these choices - because it allows you to combine both of these benefits together, giving your opponent fewer attacks and disadvantage to being hit simultaneously.

It's pretty simple too, take the Dash action each round but fall prone at the end of every turn, then stand up and Dash again at the beginning of your next turn. Let me explain:

Falling prone is free:

You can drop prone without using any of your speed

While a creature is prone:

[Unless] the attacker is within 5 feet of the creature [...] the attack roll has disadvantage.

Standing up again from prone:

costs an amount of movement equal to half your speed

When you take the Dash action, you:

gain extra movement for the current turn. The increase equals your speed, after applying any modifiers.

So, putting all of that together:

• On your first turn you take the Dash action and move 60 feet into the open.
• Then drop prone as a free action.
• On every subsequent turn you take the Dash action to double your available movement (based on your speed) from 30 ft to 60 ft.
• Standing up costs you 15 ft of movement (which is half your speed - crucially not half your movement).
• You can then run the remaining 45 ft before throwing your self prone again as a free action.

This approach gives you the best of both worlds.

Moving 60 ft in the first round and 45 ft every subsequent round will allow you to clear the field in only 6 rounds (by the 7th round, you'll have moved 330 ft and be safe and dry) - just one more than Dashing without falling prone. And, every single one of those six attacks against you will have had disadvantage.

As a minor point (credit to Michael), in a pretty small minority of cases, Dashing and remaining upright will still be the superior tactic. If the archer is practically guaranteed to hit you, as a consequence of your low AC compared to their high attack bonus, then making them roll their attacks at disadvantage will be of no real benefit to you, and you'll just want to get out of there as fast as you possibly can.

What if the attacker readies their attack to trigger once you stand up again?

This is still a win-win for you - with thanks to the commenters (Ryan C. Thompson and Anketam).

• If the attacker has extra attack or multi-attack they'll lose these additional attacks by choosing to Ready.
• If they choose a trigger of 'fire when you stand up again', you could just Dash and crawl for thirty feet and they'll have lost the attack entirely.
• Maybe they get wise to this and next time they choose a trigger of more simply 'fire when you move again' assuming that your first move will be to stand up, but allowing them an attack even if you don't. In that case simply take the Dash action and crawl forward until their triggered attack comes at you, with disadvantage, and then stand up and run afterwards, assuming it makes mathematical sense of your remaining movement to do so.
• Great suggestion. But there is an unlikely scenario where dashing all the way could still be beneficial: If your AC is extremely low or your opponent’s attack rolls are extremely good they’ll hit almost all the time. Getting across in 5 turns (dash all the way) instead of 6 turns (dash+prone) would mean one less hit. Mar 5 '20 at 9:52
• @michael bounded accuracy makes that scenario almost impossible. And if you are talking about an AC 10 character against elite snipers, he's dead anyway. Mar 5 '20 at 14:40
• Should be noted that it is almost always better to fall prone than to dodge against ranged attacks: dodge works only if you can see the attacker. Mar 5 '20 at 14:42
• In the last paragraph: Doesn't that assume you know what their trigger condition is? Mar 5 '20 at 15:24
• @MarkWells Nope, I don't think so. You can RP it as 'I'm trying to stay low until I see them take a shot, then I'll get up and run, if I think I have time to, while they're reloading. That would cover any trigger and it's not even an unrealistic way to behave under the circumstances. Mar 5 '20 at 15:38

# Depends on the probability of your enemies hitting you with an attack.

If you Dodge, the damage you take, on average, is given by the probability p of hitting you (squared because of disadvantage) multiplied by the average damage d they deal times 10 rounds times the amount of attackers a: $$\text{AvgDamage}_{\text{Dodge}} = p^2 \times d \times 10 \times a.$$

If you Dash, the damage you take, on average, is given by the probability p of hitting you multiplied by the average damage d they deal times 5 rounds times the amount of attackers a: $$\text{AvgDamage}_{\text{Dash}} = p \times d \times 5 \times a.$$

The goal is to find the breakoff point p where $$p^2 \times d \times 10 \times a = p \times d \times 5 \times a.$$ The answer is $$p = \frac{1}{2}.$$

If their probability of hitting you is above 50%, then it is better for you to Dash. If it is below 50%, it is better for you to Dodge. If it is 50%, both have the same outcome. In other words, if enemies hit you with a roll of 11, you should Dash. Otherwise, you should Dodge.

To account for critical hits (the math gets trickier, but please bear with me), you have to account for the 0.05 chance of doubling the damage dice d, but not the modifier m.

Thus, for the Dodge, the real average damage becomes $$\text{AvgDamage}_{\text{Dodge}} = ((p^2-0.05^2) \times (d+m) + 0.05^2 \times (2d+m)) \times 10 \times a.$$

For the Dash, $$\text{AvgDamage}_{\text{Dash}} = ((p-0.05) \times (d+m) + 0.05 \times (2d+m)) \times 5 \times a.$$

The breakoff point p now actually depends on the modifier m and average dice damage d. You can play around with this to find the cutoff points. For example, if the enemy rolled a d6+2 for damage, then p=0.5271. In this case, you should Dash if the enemy hits you with a 12 or higher in its die roll, otherwise Dodge. Feel free to play around with different parameters in the formula.

• It might be worthwhile to clarify that p=0.5 occurs when AC - bonus to hit equals 11 because of how the tie is resolved. It can be a bit counter-intuitive. Mar 4 '20 at 15:16
• @Medix2 I think you just need to choose a breakoff point of 47.5% instead of 50% to account for criticals (unless crits do more than double damage). Mar 4 '20 at 15:29
• @Medix2 Technically yes, the crit chance does matter for overall damage taken, but all I asked about in the question was hit chance, because factoring in crits adds a lot of complexity, since it requires you to specify the normal damage in terms of dX + Y, so you can figure out how much more damage a crit does on average. I decided that kind of complexity wasn't really what I was looking for. Mar 4 '20 at 15:47
• @Medix2 I don't think it is that simple. I've appended a critical hit formula, can you validate it? Mar 4 '20 at 21:54
• Oh that's ... Yeah that's correct, I just completely forgot damage doesn't actually double because, as you've pointed out, the modifier doesn't double along with the dice. Interesting Mar 4 '20 at 21:57

## If the attacks are likely to hit you, it's better to Dash.

The chance of a single attack roll hitting is $$\x\$$. The chance of getting hit with disadvantage on the roll is $$\x^2\$$.

If you take 2 attacks with disadvantage, the expected number of hits you'll take is $$\2x^2\$$.

So Dashing is better than Dodging when $$\2x^2 > x\$$, or $$\x > \frac{1}{2}\$$.

### Crits

The chance of a single attack roll being a crit is 1/20.

The chance of an attack with disadvantage being a crit is 1/400.

The expected number of crits you'll take from 2 attacks with disadvantage is 1/200.

All crits are also hits. The extreme case for a crit is that there's no flat modifier, only dice, and we can consider a crit to be exactly as bad as getting hit twice. Then the number of "effective hits" is just the number of expected hits + expected crits.

$$\textbf{Dash 1 round: } \quad x + \frac{1}{20}$$

$$\textbf{Dodge 2 rounds: } \quad 2x^2 + \frac{1}{200}$$

The break point is at $$\2x^2 - x - 9/200 = 0\$$, which is about $$\x = 0.54\$$. So taking crits into account, Dodging is slightly better. If you can be hit on a roll of 12, you should Dash.

This is an upper bound. The lower bound, where crits are ignored entirely, is 0.5 (as shown above). Since the difference is a little less than one step on a d20, there's no reason to worry about exactly where in that range the break point falls.

• That's an odd result in terms of simulation. I'd expect skilled attackers to be better at leading a fast but unevasive target while dodging would be less effective against less accurate fire. Mar 4 '20 at 23:45
• @smithkm The mechanics don't deal with "skill" at that level of detail. The to-hit threshold summarizes many factors including attacker skill (Dex bonus and proficiency) but also the target's agility, any armor they're wearing, and the benefit of cover. Mar 5 '20 at 15:28
• I was just saying it's interesting. I know verisimilitude and world simulation are pretty much the lowest priority in the way D&D is designed. If I were being critical I'd be going after HP and AoO not this. Mar 5 '20 at 18:53

## Heroes should dash, monsters should dodge.

• Take the dodge action each round and move 30 feet per round for 10 rounds, thus facing 10 rounds of attacks at disadvantage.
• Take the dash action and move 60 feet each round for 5 rounds, thus facing 5 rounds of attacks without disadvantage.

Which of these 2 options results in fewer attacks hitting me on average?

Other answers have already covered the "on average" part of this question, but minimising the average isn't always the best option. The spread also matters.

Scenario #1: the attackers need 11 to hit, and I have enough HP to survive at most four hits. (For simplicity's sake, ignoring variable damage.)

At 11 TH, chance of hitting is 25% with disadvantage, 50% without. So the average number of attacks hitting will be 10*0.25 = 5*0.5 = 2.5. If we only care about the average, the two options are equally good.

But what we really care about here is "do we survive?" and for that, the average doesn't tell the whole story.

If we dash (5 attacks at 50% chance to hit), there's about a 3% chance (exactly 1/32) that the archers will get lucky enough to score five hits and take us down.

But if we dodge (10 attacks at 25%), that rises to about 8%. Same average, but less consistent, means a greater chance of getting a fluke big enough to down us.

So in this case, we want to dash, to protect ourselves against the possibility of unusually bad luck.

Scenario #2: exactly the same as #1, except that this time we're already injured so we can only survive one hit.

In this case, the average outcome is very bad. But dodging results in about a 24% chance of making it, compared to only 19% for dashing. So we want to dodge, to improve the chances of the fluke we need to survive.

In a real game, all sorts of other complications apply, and the numbers will depend on the specific scenario. But in the normal course of play, encounters are balanced so that PCs will survive "average" outcomes and their enemies will not.

It then follows that as a PC, reduced randomness (spread/variance) is usually a good thing.

If the averages are very different, you're generally best off picking the option that gives the best average. But if you have two options that look similar on averages, pick the one that has less room for flukes - in this case, dashing.

OTOH, if you're an antagonist, or a PC having a very bad day, the average is not your friend. Now you're looking for a fluke, and for that you're best off increasing the spread - i.e. dodging.