Power or accuracy?

D&d (Dungeons and Dragon) is a popular RPG system usually set in the context of a medieval fantasy world. In this game, the wand that launches fireballs is a very coveted item among the characters/players, as it gives its bearer the power to shoot fireballs in the direction it points.

Thus, each shot has a precision factor that we will call x1 and a damage related to the fireball reaching its target that we will call y.

However, players have never really lived in a medieval fantasy world (note the hypocrisy …) so they think that if they can hold the wand with just one hand, they could very well use two wands and thus shoot two fireballs at the same time.

Thus, each shot of each wand has a precision factor that we will call x2 and damage related to the fireball reaching its target that we will call y.

This strategy is well accepted, the players assume that the more firepower the better, after all, if all attacks hit, they will do twice as much damage.

How much does it influence this story? Well, here’s our discussion

To start, we have to x1> x2, because the accuracy when firing two wands must be lower than when firing just one.

Thus, x2 = a1.x1 where 0 <a1 <1.

For example, if the wand accuracy was 100% and a1 was 0.8. We would have x2 = 80%.

Thus, with a wand we have a 100% chance of hitting the target and taking damage from it.

With two wands, we have a 64% chance of hitting the target with both wands and taking 2.y damage, 32% chance of hitting the target with one of the wands and taking y damage, and 4% chance of making mistakes the target with both wands and take no damage.

In this perspective, in the time needed to launch 200 fireballs, we would hit an average of 160 of them, while with just one wand we would generate the damage of 100 fireballs at the same time.

But you may be thinking that the character with 100% accuracy with a wand is too much, so let’s assume that he has 80% accuracy with a wand, that is x1 = 80% and x2 = a1.x1, with a1 = 0 , 8. That is, x2 = 64%.

In this context, with two wands we have a 41% chance of hitting the target with two fireballs at the same time, a 46% chance of hitting the target with only one of the wands and a 13% chance of not hitting the target with any wand .

In this perspective, in the time necessary to launch 200 fireballs, we would hit an average of 128, while with just one wand we would generate the damage of 80 fireballs at the same time.

But why stop there? Let’s be more daring, if with each addition of a wand we have only 80% of the previous precision, ie x1 = 0.8, x2 = (0.8) ², x3 = (0.8) ³, x4 = (0 , 8) ⁴,…, x8 = (0.8) ⁸, let’s see what happens if we start shooting with 8 wands at the same time? Is it a good deal?

In this context, we have x8 = 16.8%, something very close to the chance of 1 6-sided die falling to number 1 (16.7%).

Thus, firing the 8 wands at the same time would be something very close to rolling the 8 dice at the same time and counting how many of them fell to number 1.

The chance of not hitting any fireballs in the target in this case is 23%.

The chance of hitting a fireball at the target in this case is 37%.

The chance of hitting two fireballs on the target is 52%.

The chance of hitting three fireballs at the target is 31%.

The chance of hitting four fireballs on the target is 10%.

The chance of hitting five fireballs on the target is 2.1%.

The chance of hitting six fireballs at the target is 0.26%.

The chance of hitting seven fireballs at the target is 0.017%.

The chance of hitting eight fireballs at the target is 0.0005%.

In this perspective, in the time necessary to launch 800 fireballs, we would hit an average of 134 of them, while with just two wands we would generate the damage of 128 fireballs at the same time.

Surprising how increasing the amount of wands in this way increased on average would make the character hit just 6 more balls of fire than if he were using two wands, don’t you think?

Edited images from Parker_West, OpenClipart-Vectors and Nandin Duarte, all by Pixabay