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The Combat Drill weapon in Lancer, located on page 155 of the free version, has an unusual (at least within Lancer's system) damage-determination property that has stymied my attempts to calculate its average damage and heat.

For those unaware, the weapon, when rolling for damage, rerolls 1s as per its Overkill tag, dealing 1 heat (internal damage to a mech) to the user for each reroll. When the Combat Drill hits a target that is Immobilized, Prone, or Stunned and its Overkill tag is triggered (i.e. one of its dice rolls a 1), it deals an additional d6 of bonus damage. Damage dice that have already triggered this property can do so again and dice generated by this property can trigger it, theoretically leading to an exponentially increasing number of d6s of damage.

The drill deals 3d6 Kinetic damage and 1d6 Energy damage by default, but bonus damage is a type of the user's choice from among already-dealt damage. How much damage is dealt on average by the drill when its secondary property is triggered? How much heat does the user take on average when the property is triggered?

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    \$\begingroup\$ For your (and for potential answerers’) awareness, the property of having dice that have a chance to add another die (that have a chance to add another die, etc.) is known as “exploding” dice. It’s not used much in Lancer but in some games it’s the default, e.g. for critical hits or whatever. I don’t, off the top of my head, know how to calculate the expected damage from exploding dice, but it might be something you (or another answerer) could research. (The Combat Drill is still somewhat unusual in that it triggers a reroll at the same time it “explodes.”) \$\endgroup\$ – KRyan May 13 at 18:57
  • \$\begingroup\$ Related, a bit: "How does the Overkill weapon tag interact with critical hits?" \$\endgroup\$ – Medix2 May 13 at 19:06
  • \$\begingroup\$ Welcome to the site! Take the tour when you have the time. \$\endgroup\$ – Glazius May 14 at 5:19
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So you have two infinite series. No big.

The expected damage value of an overkill d6 is 4. It can only come up 2 through 6 with equal probability, so just take the average of those. The expected heat value of an exploding overkill d6 is 1/6, plus 1/6 of 1/6, and so on, which may be familiar as an infinite geometric series (for |r| < 1):

$$ \sum\limits_{k = 0}^\infty {ar^{k} = \frac{a}{{1 - r}}} $$

For a and r both = 1/6, the series sums to 1/5, so the expected heat value of an overkill d6 is 1/5.

For the combat drill's special overkill property, the expected number of additional dice per overkill d6 is also 1/5, but each of those is an overkill d6, and so on, and so on.

Since the combat drill rolls 4d6, a = 4 and r = 1/5, and the expected number of overkill d6s is$$\frac{4}{{1 - \frac{1}{5}}} = \frac{4}{\frac{4}{5}} = 5$$

Rolling 5 overkill d6s is expected to deal 20 damage and generate 1 heat.

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