Editing Item drops

Items that drop from [[monsters]] or [[item boxes]] all follow a common set of drop rate calculations that determine which item drops, if one drops at all. Items obtained via other methods, such as gambling {{Meseta}} with [[Coren]] or redeeming [[Lucky Coins]] in {{Quest link|Gallon's Shop}}, use their own unique sets of calculations. If chance is involved in obtaining items, the mechanism by which the drop is determined is commonly referred to as a '''roll.'''

==Item rolls==
A roll can be envisioned as rolling a many-sided die, where rolling a 1 means the item will drop. For example, if a monster drops an item with a <code>1/200</code> drop rate, it is equivalent to rolling a 200-sided die.

===Monsters===

Each time a monster is defeated, the game rolls to determine whether an item will drop, and if so, what that item will be, in the following order:

#The game rolls for event drops, such as {{Tool|rare|Event Egg}}, {{Tool|rare|Halloween Cookie}}, or {{Tool|Present (Christmas)}}
#The game rolls for [[music disks]] (<code>1/600</code> from any monster)
#The game rolls based on the monster's [[drop anything rate]] (DAR) to determine if the monster has the possibility of dropping anything else
#The game rolls to decide if the monster will drop its specified rare item (depending on area, section ID, and difficulty)
#The game rolls to determine whether the drop is {{Meseta}}, a [[tools|tool]], or the monster's set drop. Each of these categories has an equivalent <code>1/3</code> chance of being rolled

====Drop rate boosts====

The chance to obtain items from monsters can be increased with drop rate boosts. [[drop anything rate|DAR]] boosts increase the chance to obtain any item at all from monsters, which can also help with its designated rare drop. [[Rare drop rate]] (RDR) boosts only increase the chance for the designated rare drop. However, a monster can't have more than 100% DAR. Certain monsters already have 100% (bosses, rare monsters) or otherwise high DAR, meaning strong DAR boosts like the 25% weekly boost won't have their full effect. A similar rule applies to RDR, in which the maximum RDR can only be boosted to <code>7/8 (87.50%)</code>. This, however, applies to very few monsters (e.g. rare Rappies from VR Temple). 

The [[drop charts]] show the combination of a monster's DAR and RDR. Individual DAR and RDR values are visible by hovering with a mouse over the drop rate or by tapping on them on mobile devices.

====Set drops====

A monster's '''set drop''' is the type of item that it will drop if it fails its rare item roll and doesn't drop either Meseta or a tool. A set drop is a category of item, not a specific one; for example, a [[Bartle]]'s set drop is a weapon, so it can drop any weapon that would be able to be dropped in the area and section ID it appears in. A set drop can be a [[Weapons|Weapon]], a [[Frames|Frame]], a [[Barriers|Barrier]], a [[Units|Unit]], or nothing.

====Rare monsters====

Some monsters can spawn as a rare variant. The spawn rate is <code>1/500</code>, with the exception of the Episode 4 boss [[Kondrieu]] spawning at a rate of <code>1/10</code>. Rare monsters have 100% DAR and a very high RDR. The overall chance to obtain an item from a rare monster can be calculated by combining the spawn rate with the RDR. For instance, on Episode 1 Ultimate, [[Hildetorr|Hildetorrs]] spawn with a rate of <code>1/500</code> and drop a [[Heaven Punisher]] on {{Bluefull}} ID with a drop rate of <code>1/204.8</code>. The overall chance to obtain this item from a [[Hildelt]] thus is <code>1/102,400</code>. The weekly Rare Monster boost increases the spawn rate by 50%.

===Item box===

Some item boxes may be set to drop specific items (e.g. some boxes in {{Quest link|1-1:Planet Ragol}} will only ever drop {{Tool|Monomate|Monomates}}), or specific ''categories'' of items (e.g. some boxes in {{Quest link|1-3:Subterranean Den}} will only ever drop tools).

For non-drop-specific item boxes, each time an [[Item box#Item box drop tables|item box]] is opened, the game rolls to determine whether an item will drop, and if so, what that item will be in the following order:

#The game rolls to decide if the box will drop its specified rare items (depending on area, section ID, and difficulty). In many areas, item boxes have multiple rare drops assigned to them. The roll order is from top to bottom in the [[drop charts]]
#The game rolls to determine whether the drop is a [[weapon]], [[frames|frame]], [[barriers|barrier]], [[units|unit]], [[tools|tool]], {{Meseta}}, or nothing (depending on area and difficulty; {{Yellowboze}} has a higher chance to obtain Meseta from item boxes).

Item box drops '''are not affected''' by '''any''' drop rate bonuses, including (but not limited to): 
* [[drop anything rate]] (DAR); 
* [[rare drop rate]] (RDR); 
* [[Anguish]] common weapon rates; 
* {{Tool|rare|Photon Drop}} drop bonuses; 
* event drop rate bonuses;
* altered weapon percentage [[pattern|patterns]] from [[Anguish]]

===Calculation===
The overall chance to obtain an item is best explained in an example. Assuming the [[Easter event]] is currently active, {{Tool|rare|Event Egg|Event Eggs}} can be obtained by killing monsters. [[Claw]]s have weapons as a set drop. The calculation of receiving a weapon box from killing a Claw on Ultimate is executed by following the roll order from above:

* Roll for Event Egg fails: <code>1 - 1/500 = '''499/500'''</code>
* Roll for music disk fails: <code>1 - 1/600 = '''599/600'''</code>
* DAR roll successful: <code>'''3/10''' (30%)</code>
* RDR roll fails: <code>1 - 1/64 = '''63/64'''</code>
* Roll for set drop successful: <code>'''1/3'''</code>

The overall chance can now be calculated by multiplying the above rates: 

<code>499/500 * 599/600 * 3/10 * 63/64 * 1/3 = '''1/10.89 (9.18%)'''</code>

From this point, further calculations can be performed in a similar fashion to determine more specific drop rates, e.g. which weapon will drop and with what attributes (see [[weapon attribute drop tables]] for further information). 

====Multiple rolls====
The overall chance of obtaining items by killing several monsters or destroying several boxes can also be calculated. For multiple rolls, the overall chance to receive a drop naturally increases. However, this increase is not linear. The increase is determined by following the binomial distribution formula:

<code>''chance of at least'' 1 ''drop'' = 1 - (1 - ''drop rate'') ^ (''number of events'')</code>

Using the above example, the overall chance of receiving at least 1 weapon box from a Claw by killing it '''10''' times is:

<code>''chance of at least'' 1 ''drop'' = 1 - (1 - ''1/10.89'') ^ (''10'') = '''1/1.62 (61.83%)'''</code>

Killing more enemies naturally increases the likelihood of receiving an item, but it will never be 100% guaranteed. This is contrary to the popular belief that killing a monster <code>X</code> times for its <code>1/X</code> item drop will give you a guaranteed drop. In reality, the overall chance in this case always comes out to roughly <code>'''63%'''</code> (approaching <code>1 - 1/e</code>, where e = Euler's number). This is statistically defined as the average chance where at least 1 drop will occur, and is particularly useful when performing calculations such as "how many runs will it take to get this item?"

=====Combined drop rates=====
Some monsters drop the same item but with different drop rates. For a drop rate <code>A</code> and a drop rate <code>B</code> the combined drop rate <code>C</code> is:

<code>C = A + B - A * B</code>

For instance, if one enemy drops an item with a drop rate of <code>1/2</code> and another with a drop rate of <code>1/4</code>, the overall chance of receiving at least 1 drop from killing both monsters is:

<code>1/2 + 1/4 - 1/2 * 1/4 = '''1/1.6 (62.5%)'''</code>

====Multiple drops====
The numbers above show the process of obtaining at least 1 item, which is sufficient for most players who hunt for a certain drop once. The chance to obtain multiple items can be calculated too. For example, if a monster drops an item with a drop rate of <code>1/200</code> and is killed 100 times, the chance of it dropping <code>3</code> items or more can be calculated:

<code>''chance of at least'' 3 ''drops'' = 1 - P(0) - P(1) - P(2)</code>, where <code>P(k)</code> is the chance of ''exactly'' <code>k</code> item drops.

So, in this example, the chance of receiving exactly 0, 1 or 2 drops has to be calculated. The general formula for <code>P(k)</code> is:

<code>P(k) = n_C_k * p^k * (1-p)^(n-k)</code>, where:

* n: number of trials, in this example '''100''' monster kills
* k: number of successes, which for P(0) is '''0''', P(1) '''1''', etc.
* p: probability of success, which is the drop rate of '''1/200'''
* n_C_k: the binomial coefficient, which is defined as '''n!/(k!(n-k)!)''' with "!" referring to the factorial

The process of evaluating each <code>P(k)</code> value is the same, it is shown here for <code>P(2)</code>:
* n = '''100'''
* k = '''2'''
* p = '''1/200'''
* n_C_k = 100!/(2!(100-2)!) = '''4950'''

<code>P(2) = 4950 * (1/200)^2 * (1-1/200)^(100-2) = '''1/13.21'''</code>

Repeating the process for <code>P(0)</code> and <code>P(1)</code> we arrive at:

<code>''chance of at least'' 3 ''drops'' = 1 - 1/1.65 - 1/3.29 - 1/13.21 = '''1/70.91 (1.41%)'''</code>

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