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Difference between revisions of "Critical hit"
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In aethership vs. denizen combat, crew members can score critical hits while manning a battle turret. The chance of critical hits there is determined by one's skill in [[Aethercraft]] rather than level, however. | In aethership vs. denizen combat, crew members can score critical hits while manning a battle turret. The chance of critical hits there is determined by one's skill in [[Aethercraft]] rather than level, however. | ||
Note that certain denizens are immune to critical hits - notably city and commune guards and certain extraordinarily powerful denizens (i.e., the Demon | Note that certain denizens are immune to critical hits - notably city and commune guards and certain extraordinarily powerful denizens (i.e., the [[Demon Lord]]s of [[Nil]]). | ||
< | == Maths == | ||
=== Expected Value === | |||
To calculate the expected value (aka the mean) of some attack a:<br> | |||
Let d = the base damage of the attack<br> | |||
Let c = the percentage critical hit chance | |||
{| class="wikitable" | |||
! Type | |||
! Damage | |||
! Chance | |||
! Expected Value | |||
|+Expected Value Calculation | |||
|- | |||
|Normal hit||d||1 - c||d(1-c) | |||
|- | |||
|CRITICAL||2 * d||c - c^2||2d(c-c^2) | |||
|- | |||
|CRUSHING CRITICAL||4 * d||c^2 - c^4||4d(c^2-c^4) | |||
|- | |||
|OBLITERATING CRITICAL||8 * d||c^4 - c^8||8d(c^4-c^8) | |||
|- | |||
|ANNIHILATINGLY POWERFUL CRITICAL||16 * d||c^8 - c^16||16d(c^8-c^16) | |||
|- | |||
|WORLD-SHATTERING CRITICAL||32 * d||c^16||32dc^16 | |||
|} | |||
So our expected value for this attack is the sum of rightmost column above<br> | |||
d(1-c) + 2d(c-c^2) + 4d(c^2-c^4) + 8d(c^4-c^8) + 16d(c^8-c^16) + 32dc^16<br> | |||
Which expands to<br> | |||
16dc^16 + 8dc^8 + 4dc^4 + 2dc^2 + dc + d<br> | |||
and we can factor this into<br> | |||
d((4c^8+1)^2 + 4c^4 + 2c^2 + c) | |||
[[Category:Combat]] | |||