OoT Rando Probability Discussion

OoT Rando Probability Discussion:

Analysis of required items vs probability of checks.  Also with and without keysanity and maps and compasses.  Let me know if any of my math or assumptions is far off and I'll correct it.  If anyone else has written about randomizer logic, I'd be interested to read it.

The math done here is rough math and does not address slight skews due to "Randomizer logic" and certain dungeons (Shadow, Water, Ice Cavern etc) not being required in some uncommon cases.  I'd emphasize that randomizer logic HARDLY skews items.  Probably less than a 1% probability gain in most cases.  An arbitrary required item has "practically" equal probability of being in any of the locations you can currently reach.  Once something is received that changes the search space, once again any arbitrary required item has practically equal probability of being in any of the locations you can now reach.  There are some cases when you can make specific conclusions based on the logic.  Other than specific things dictated specifically by the logic, that logic should most often not be extended to imply that one place is any more likely than another to have the thing you need at a given point in time.

Summary: Something you need is pretty much randomly distributed somewhere you can get to it, within the logic.

Probability Discussion:

Note: This assumes you're playing with Ganon's trials on/required.

Assume the following are required items.  (They are each needed well over 50% of the time.)  (Not counting songs or medallions)
2 of 2 hookshots.
3 of 3 progressive strength upgrades.
Hammer
Bottle with Letter
Boomerang
Dins Fire
Light Arrows
Iron Boots
Hover Boots
Mirror Shield

13 items are required for the majority of runs.  Let's use this 13 number for our analysis.  Since we're making these assumptions, the math is an estimation.

The following I did not count, most of the time all of the multiples are not blocked, and others aren't needed that often:
1 of 3 slingshots.
1 of 3 bomb bags.
1 of 3 bows.
1 of 2 magic meters.
Lens of Truth
Progressive Scales (x2),
Fire Arrows,
Kokiri Sword

First, analyzing with no keysanity, no randomized maps and compasses.

In 2.0 with no keysanity, with heart containers, heart pieces and grottos randomized, there are 176 base checks, 10 maps and 10 compasses in dungeons.  Let's subtract the 19 chests in ganon's castle.  You wouldn't get these 13 items in there most of the time.  So use 157 checks as our estimation.  A given chest has a 13/157 (8.2%) chance of having something you absolutely need.  There is also a smaller chance it has something less important you absolutely need, so in general, these estimates will be slightly lower than actual (probably within a few percentage points.)

Approximate chance you need at least N skulltulas.  b(x; n,P)
https://stattrek.com/online-calculator/binomial.aspx
10 skulltulas: 34.8%   b(x>=1; 5, 0.082)
20 skulltulas: 28.9%   b(x>=1; 4, 0.082)
30 skulltulas: 22.6%   b(x>=1; 3, 0.082)
40 skulltulas: 15.7%   b(x>=1; 2, 0.082)
50 skulltulas: 8.2% (13/157)   b(x>=1; 1, 0.082)

This line thought can also be extended to the following statement: There is about a 34.8% chance you will have to check the 5th worst check you don't know anything about (Accounting for stone of agony information for example.)

N=5 34.8%
N=6 40.2%
N=7 45.1%
N=8 49.6%
N=9 53.7%
N=10 57.5%
N=11 61.0%
N=12 64.2%  (Math: 157/13 is ~12,  1 - 1/e is 63.2%.)
N=13 67.1%
N=14 69.8%
N=15 72.3%
N=16 74.6%
N=17 76.6%
N=18 78.6%
N=19 80.3%
N=20 81.9%

This math fluctuates as you do more checks.  For example if you need 3 more items and there are only 40 checks remaining, you can replace the math with the current state to get an accurate probability current to your state.

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Switch maps and compasses out for rupees and randomize the extra dungeon chests:
196 base checks. Subtract 19 in Ganon's Castle
A given chest has a 13/177 (7.3%) chance of having something you absolutely need.

Approximate chance you need at least N skulltulas.  b(x; n,P)
10 skulltulas: 31.5%   b(x>=1; 5, 0.073)
20 skulltulas: 26.1%   b(x>=1; 4, 0.073)
30 skulltulas: 20.3%   b(x>=1; 3, 0.073)
40 skulltulas: 14.1%   b(x>=1; 2, 0.073)
50 skulltulas: 7.3% (13/157)   b(x>=1; 1, 0.073)

Example: If you turn on maps and compasses, you will have to do 50 skulltulas about 1 less time in 100 seeds.  Have you actually finished 100 seeds yet?

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Now let's do the same math for keysanity.  First no maps and compasses.

Here's my count on minimum required keys to beat the boss of a dungeon, I might be wrong:
Forest Temple: 5/5 keys
Fire Temple: 0/8 keys
Water Temple: 0/6 keys
Shadow: 5/5 keys
Spirit: 3/5 keys adult side   (2 more of the 5 to do child side)
Ganon's Tower 2/2 keys:
GTG: 0/9 keys
Bottom of the Well: 0/3 keys
Total Boss keys: 6

49 keys and boss keys.

In 2.0 with keysanity, with heart containers, heart pieces and grottos randomized, there are 225 base checks, 10 maps and 10 compasses in dungeons.  Let's subtract the 19 chests in ganon's castle.  You wouldn't get most of these items in there the majority of the time.  So use 206 checks as our estimation.  Now we've introduced 6 boss keys and 14 keys that we mostly need all of a majority of the time.

A given chest has a 32/206 (15.5%) chance of having something you absolutely need.

Approximate chance you need at least N skulltulas.  b(x; n,P)
10 skulltulas: 56.9%   b(x>=1; 5, 0.155)
20 skulltulas: 49.0%   b(x>=1; 4, 0.155)
30 skulltulas: 39.7%   b(x>=1; 3, 0.155)
40 skulltulas: 28.6%   b(x>=1; 2, 0.155)
50 skulltulas: 15.5%   b(x>=1; 1, 0.155)

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Now with maps and compasses.

Switch maps and compasses out for rupees and randomize the extra dungeon chests:
245 base checks. Subtract 19 in Ganon's Castle
A given chest has a 32/226 (14.2%) chance of having something you absolutely need.

Approximate chance you need at least N skulltulas.  b(x; n,P)
10 skulltulas: 53.5%   b(x>=1; 5, 0.142)
20 skulltulas: 45.8%   b(x>=1; 4, 0.142)
30 skulltulas: 36.8%   b(x>=1; 3, 0.142)
40 skulltulas: 26.4%   b(x>=1; 2, 0.142)
50 skulltulas: 14.2%   b(x>=1; 1, 0.142)

Example: If you turn on maps and compasses, you will have to do 50 skulltulas about 1 less time in 100 seeds.  Have you actually finished 100 seeds yet?

Since it seems like people are playing a lot of keysanity with no maps and compasses, let's do the rest of the math:

N=5 53.5%
N=6 60.1%
N=7 65.8% (Math: 226/32 is ~7,  1 - 1/e is 63.2%.)
N=8 70.6%
N=9 74.8%
N=10 78.4%
N=11 81.4%
N=12 84.1%
N=13 86.3%
N=14 88.3%
N=15 89.9%

This line thought can also be extended to the following statement: There is about a 52.1% chance you will have to check the 5th worst check you don't know anything about (Such as stone of agony information.)

This math fluctuates as you do more checks.  For example if you need 3 more items and there are only 40 checks remaining, you can replace the math with the current state to get an accurate probability current to your state.

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In conclusion,

If I was playing randomizer, with keysanity, with maps and compasses randomized, I would play with N=5 52.1% in mind.  There is a 52.1% chance that if I check all stones of agony, I will have to check all but the worst 5 checks that are not stone of agony checks.  A couple really bad checks that come to mind are skulltulas and the chests deep in the water temple.  If the seed is going well, or if some of the adult dungeons aren't required to be beaten, we can raise N=5 towards maybe N=8.  At the end of the day you can't skip much and come out ahead on average.