Random list mechanism

There are many things in the game that follow this mechanism to choose different effect/type randomly such as Golden cookie, Wrath cookie, Force the Hand of Fate, Garden mutations, and Sugar Lump types. Basically the mechanism starts with a pool which contains certain effects. Then it adds different effects to the pool randomly. Finally it chooses one effect from the pool with equal probability.

Garden Mutations

At each tick, the game will check every empty plots for possible mutation recipes. If certain conditions are satisfied, there will be a corresponding base chance to add the mutated plant into the candidate pool. The final outcome will be chosen from the candidate pool with equal probability. If the soil is Woodchips, the above procedure will loop 3 times at each tick.

For example, if an empty plot is surrounded by 8 Queenbeets, there are 3 possible mutations with an equal base chance of p=0.001; Juicy Queenbeets, Duketaters, and Shriekbulbs. The candidate pool has 2^3=8 possibilities and can be grouped into four kinds according to the size. All pools in the same group has the same probability to be generated.

Group ID	Pool size	Pool generation probability	Number of pools
G0	0	(1-p)^3	1
G1	1	p(1-p)^2	3
G2	2	p^2(1-p)	3
G3	3	p^3	1

Among the all eight pools, there are four pools contains Shriekbulbs: 1, 2 and 1 belong to G1, G2 and G3, respectively. The probability of picking Shriekbulbs in pool Gi will be 1/i. After adding all four pools together, the total probability of picking Shriekbulbs is P = p(1-p)^2 + 2*p^2(1-p)/2 + p^3/3 = p - p^2 + p^3/3 = 0.000999. Since in this case all three mutating plants have the same base chance, the probability of picking Duketaters and Juicy Queenbeets will also be P = p - p^2 + p^3/3. Hence the probability of picking nothing P₀ = 1 - 3P.

If the soil is Woodchips, the probability of mutating Shriekbulbs after three loops will be W = P + P*P₀ + P*P₀^2 = 3p - 12p^2 + 28p^3 - 42p^4 + 42p^5 - 28p^6 + 12p^7 - 3p^8 + p^9/3 = 0.002997. Hence the probability of picking nothing W₀ = 1 - 3W.

Take another example, if an empty plot is surrounded by 2 Baker's Wheat, then there are three mutation candidates; Baker's Wheat, Thumbcorn and Bakeberry. These have a base chance of p₁=0.2, p₂=0.05 and p₃=0.001, respectively. There are 2^3=8 possible candidate pools.

Candidate pool contents			Probability
Baker's Wheat	Thumbcorn	Bakeberry
Yes	Yes	Yes	p1p2p3
Yes	Yes	No	p1p2(1-p3)
Yes	No	Yes	p1(1-p2)p3
No	Yes	Yes	(1-p1)p2p3
Yes	No	No	p1(1-p2)(1-p3)
No	Yes	No	(1-p1)p2(1-p3)
No	No	Yes	(1-p1)(1-p2)p3
No	No	No	(1-p1)(1-p2)(1-p3)

The probability of picking Baker's Wheat, Thumbcorn, Bakeberry or nothing are:

• Baker's Wheat: P₁ = p₁p₂p₃/3 + p₁p₂(1-p₃)/2 +p₁(1-p₂)p₃/2 + p₁(1-p₂)(1-p₃) = 0.194903
• Thumbcorn: P₂ = p₁p₂p₃/3 + p₁p₂(1-p₃)/2 + (1-p₁)p₂p₃/2 + (1-p₁)p₂(1-p₃) = 0.0449783
• Bakeberry: P₃ = p₁p₂p₃/3 + (1-p₁)p₂p₃/2 + p₁(1-p₂)p₃/2 + (1-p₁)(1-p₂)p₃ = 0.000878333
• Nothing: P₀ = (1-p₁)(1-p₂)(1-p₃) = 1 - P₁ - P₂ - P₃ = 0.75924

If the soil is Woodchips, the probability of picking Baker's Wheat, Thumbcorn, Bakeberry or nothing are:

• Baker's Wheat: W₁ = P₁+ P₁P₀ + P₁P₀^2 = 0.455233
• Thumbcorn: W₂ = P₂ + P₂P₀ + P₂P₀^2 = 0.105055
• Bakeberry: W₃ = P₃ + P₃P₀ + P₃P₀^2 = 0.00205151
• Nothing: W₀ = P₀^3 = 1 - W₁ - W₂ - W₃ = 0.43766

Force the Hand of Fate

The procedure of Force the Hand of Fate is a little different. For golden cookie, the procedure is:

Win:

• Add Frenzy and Lucky to a pool.
• If there is no Dragonflight buff active, add Click Frenzy.
• 10% chance to add Storm, Storm and Blab to the pool.
• 25% chance to add Building Special, if the you owned more than or equal to 10 total building.
  • For all building of which number more or equal to 10, choose one and buff according to the chosen building's amount. If no building has such quantity, choose Frenzy.
• 15% chance to replace the pool with Storm Drop.
• 0.01% chance to add Free Sugar Lump to the pool.
• Pick a random effect from the list.

Note the there is 15% chance that the pool is replaced completely instead of adding something into the pool. To handle this rule properly, consider two cases. First assume that the pool is never replaced.

Probability without Dragonflight (Win)
Frenzy=Lucky=C. Frenzy	Storm	Blab	Building	Sugar
0.2973144554	0.03214241071	0.01607120536	0.05982025893	0.00002275892857

Second, assume that the pool is always replaced:

Probability without Dragonflight (Win)
StormDrop	Sugar
0.99995	0.00005

The actual probability will be mix of these two cases.

Probability without Dragonflight (Win)
Frenzy=Lucky=C. Frenzy	Storm	Blab	Building	StormDrop	Sugar
0.2527172871	0.02732104911	0.01366052455	0.05084722009	0.1499925	0.00002684508929

Fail:

• Add Clot and Ruin to a pool.
• 10% chance to add Cursed Finger, Elder Frenzy to the pool.
• 0.3% chance to add Free Sugar Lump to the pool.
• 10% chance to replace the pool with Blab.
• Pick a random effect from the list.

Probability (Fail)
Clot=Ruin	Cursed Finger=Elder Frenzy	Sugar	Blab
0.4270815	0.0224865	0.000864	0.1

Golden/Wrath cookie

For Golden cookie and Wrath cookie, there is an additional rule. Take the procedure of picking effect of golden cookie in version 1.0466 as an example:

• Add Frenzy and Lucky to a pool.
• 3% chance to add Chain to the pool, if at least 100,000 cookies have been baked in this game.
• 10% chance to add Click Frenzy to the pool.
• 80% chance to remove the last effect from the pool, if it's there.
• 0.01% chance to add Blab to the pool.
• Pick a random effect from the pool.

The rule 5 "80% chance to remove the last effect from the pool, if it's there." makes things more complicated but still doable. First we shall ignore the rule first, and calculate the probability by listing all possible pools as mentioned above. (From now on we shall assume that at least 100,000 cookies have been baked.)

Pool				Probability
# of effect	Chain	C. Frenzy	Blab
5	X	X	X	P1=.03*.1*.0001
4	X	X		P2=.03*.1*.9999
4	X		X	P3=.03*.9*.0001
3	X			P4=.03*.9*.9999
4		X	X	P5=.97*.1*.0001
3		X		P6=.97*.1*.9999
3			X	P7=.97*.9*.0001
2				P8=.97*.9*.9999

Therefore, the probability of picking Frenzy is: P1/5+(P2+P3+P5)/4+(P4+P6+P7)/3+P8/2=0.478568. The probability of picking Lucky is the same as Frenzy. The probability of picking Chain,Click Frenzy and Blab are P5/5+(P2+P3)/4+P4/3=0.00974976, P5/5+(P2+P5)/4+P6/3=0.0330825 and P5/5+(P3+P5)/4+P7/3=0.00003226 respectively. These probabilities are called primitive probabilities.

Primitive probability
Frenzy	Lucky	Chain	C. Fenzy	Blab
0.478568	0.478568	0.00974976	0.033082	0.00003226

However, due to the rule 5, the actually probability is depend on last cookie. For example, if the last cookie is Frenzy, there are 80% chance that we shall apply the following table.

Pool				Probability
# of effect	Chain	C. Frenzy	Blab
4	X	X	X	P1=.03*.1*.0001
3	X	X		P2=.03*.1*.9999
3	X		X	P3=.03*.9*.0001
2	X			P4=.03*.9*.9999
3		X	X	P5=.97*.1*.0001
2		X		P6=.97*.1*.9999
2			X	P7=.97*.9*.0001
1				P8=.97*.9*.9999

From the table, we can calculate the conditional probability:

	Conditional probability
Last Cookie	Frenzy	Lucky	Chain	C. Frenzy	Blab
Frenzy	0	0.935954	0.0144995	0.0494984	0.0000478583

Remember that the actual probability should be a combination of 20% of primitive probability and 80% of conditional probability:

	Actual probability
Last Cookie	Frenzy	Lucky	Chain	C. Frenzy	Blab
Frenzy	0.0957135	0.844477	0.0135496	0.0494984	0.00004783

The actual probability of Lucky is the same as Frenzy. For conditional probability of Chain, we shall consider the following table:

Pool				Probability
# of effect	Chain	C. Frenzy	Blab
4	X	X	X	P1=.03*.1*.0001
3	X	X		P2=.03*.1*.9999
3	X		X	P3=.03*.9*.0001
2	X			P4=.03*.9*.9999
4		X	X	P5=.97*.1*.0001
3		X		P6=.97*.1*.9999
3			X	P7=.97*.9*.0001
2				P8=.97*.9*.9999

And the actual probability is:

	Actual probability
Last Cookie	Frenzy	Lucky	Chain	C. Frenzy	Blab
Chain	0.482368	0.482368	0.00194995	0.0332825	0.000032452

Repeating the process, we can build up the following table. Notice that since we add Blab after rule 5, the conditional/actual probability when last cookie is blab is the same as primitive probability.

	Current Cookie
Last Cookie	Frenzy	Lucky	Chain	Click Frenzy	Blab
Frenzy	.095713547	.8444769536	.013549572	.0462151886	.0000447386
Lucky	.8444769536	.095713547	.013549572	.0462151886	.0000447386
Chain	.482367547	.482367547	.001949952	.033282502	.000032452
Click Frenzy	.4917004136	.4917004136	.009949752	.006616502	.0000329186
Blab/None	.478567735	.478567735	.00974976	.03308251	.00003226

The table above is a transition matrix, the long term (stationary) probability can be found as the eigenvector of the transition matrix with eigenvalue 1.

The Long Term Probability
Frenzy	Lucky	Chain	Click Frenzy	Blab
47.121495%	47.121495%	1.3236432%	4.4289617%	0.0044052%

Using this, we can obtain the following table shows the final result:

	Current Cookie
Last Cookie	Frenzy	Lucky	Chain	Click Frenzy	Blab	Total
Frenzy	4.5101654%	39.793016%	0.6384761%	2.1777288%	0.0021082%	47.121495%
Lucky	39.793016%	4.5101654%	0.6384761%	2.1777288%	0.0021082%	47.121495%
Chain	0.6384825%	0.6384825%	0.0025810%	0.0440542%	0.0000430%	1.3236432%
Click Frenzy	2.1777223%	2.1777223%	0.0440671%	0.0293042%	0.0001458%	4.4289617%
Blab	0.0021082%	0.0021082%	0.0000429%	0.0001457%	0.0000001%	0.0044052%
Total	47.121495%	47.121495%	1.3236432%	4.4289617%	0.0044052%	100%

For wrath cookies, the procedure of determining effect is different:

• Add Lucky, Clot and Ruin to a pool.
• 30% chance to add Elder Frenzy and Chain to the pool.
• If previous step fails (70%), 3% chance to add Chain to the pool, if at least 100,000 cookies have been baked this game.
• 10% chance to add Click Frenzy to the pool.
• 80% chance to remove the last effect from the pool, if it's there.
• 0.01% chance to add Blab to the pool.
• Pick a random effect from the pool.

However, the transition matrix can (assuming at least 100,000 cookies have been baked) still be made:

	Current Cookie
Last Cookie	Lucky	Clot	Ruin	E.Frenzy	Chain	C. Frenzy	Blab
Lucky	.056962699	.383759471	.383759471	.070598646	.077087491	.027805042	.000027181
Clot	.383759471	.056962699	.383759471	.070598646	.077087491	.027805042	.000027181
Ruin	.383759471	.383759471	.056962699	.070598646	.077087491	.027805042	.000027181
E. Frenzy	.296413112	.296413112	.296413112	.011799806	.075743544	.023194559	.000022754
Chain	.297757059	.297757059	.297757059	.070598646	.012828785	.023278556	.000022835
C. Frenzy	.290223955	.290223955	.290223955	.059799006	.065027901	.004478916	.000022311
Blab	.284813495	.284813495	.284813495	.058999029	.064143927	.022394582	.000021977

And the long term probability can be calculated:

The Long Term Probability
Lucky	Clot	Ruin	E. Frenzy	Chain	Click Frenzy	Blab
27.832181%	27.832181%	27.832181%	6.6406924%	7.2047971%	2.6553223%	0.0026445%

The final table then can be obtained:

	Current
Last	Lucky	Clot	Ruin	E. Frenzy	Chain	C. Frenzy	Blab
Lucky	1.5853962%	10.680863%	10.680863%	1.9649143%	2.1455130%	0.7738750%	0.0007565%
Clot	10.680863%	1.5853962%	10.680863%	1.9649143%	2.1455130%	0.7738750%	0.0007565%
Ruin	10.680863%	10.680863%	1.5853962%	1.9649143%	2.1455130%	0.7738750%	0.0007565%
E.Frenzy	1.9683883%	1.9683883%	1.9683883%	0.0783589%	0.5029896%	0.1540279%	0.0001511%
Chain	2.1452792%	2.1452792%	2.1452792%	0.5086489%	0.0924288%	0.1677173%	0.0001645%
C.Frenzy	0.7706381%	0.7706381%	0.7706381%	0.1587856%	0.1726700%	0.0118930%	0.0000592%
Blab	0.0007532%	0.0007532%	0.0007532%	0.0001560%	0.0001696%	0.0000592%	0.0000001%
Total	27.832181%	27.832181%	27.832181%	6.6406924%	7.2047971%	2.6553223%	0.0026445%

Grandmapocalypse

There are four phases in Grandmapocalypse: Appeased, Awoken, Displeased, Angered. For the first phase all the cookies will be golden and for the last phased all cookies will be wrath.

In two middle phases, the probability of wrath cookie is 1/3 and 2/3 respectively. Thus the transition matrix of these two phase are a linear combination of transition matrix of Appeased phase (pure golden cookie) and transition matrix of Awoken phase (pure wrath cookie).

Let us show the golden transition matrix and wrath transition matrix first:

Golden	Current Cookie
Last Cookie	Frenzy	Lucky	Clot	Ruin	E. Frenzy	Chain	C. Frenzy	Blab
Frenzy	.095713547	.8444769536	0	0	0	.013549572	.0462151886	.0000447386
Lucky	.8444769536	.095713547	0	0	0	.013549572	.0462151886	.0000447386
Clot/None	.478567735	.478567735	0	0	0	.00974976	.03308251	.00003226
Ruin/None	.478567735	.478567735	0	0	0	.00974976	.03308251	.00003226
E.Frenzy/None	.478567735	.478567735	0	0	0	.00974976	.03308251	.00003226
Chain	.482367547	.482367547	0	0	0	.001949952	.033282502	.000032452
C.Frenzy	.4917004136	.4917004136	0	0	0	.009949752	.006616502	.0000329186
Blab/None	.478567735	.478567735	0	0	0	.00974976	.03308251	.00003226

Wrath	Current Cookie
Last Cookie	Frenzy	Lucky	Clot	Ruin	E. Frenzy	Chain	C. Frenzy	Blab
Frenzy /none	0	.284813495	.284813495	.284813495	.058999029	.064143927	.022394582	.000021977
Lucky	0	.056962699	.383759471	.383759471	.070598646	.077087491	.027805042	.000027181
Clot	0	.383759471	.056962699	.383759471	.070598646	.077087491	.027805042	.000027181
Ruin	0	.383759471	.383759471	.056962699	.070598646	.077087491	.027805042	.000027181
E.Frenzy	0	.296413112	.296413112	.296413112	.011799806	.075743544	.023194559	.000022754
Chain	0	.297757059	.297757059	.297757059	.070598646	.012828785	.023278556	.000022835
C.Frenzy	0	.290223955	.290223955	.290223955	.059799006	.065027901	.004478916	.000022311
Blab	0	.284813495	.284813495	.284813495	.058999029	.064143927	.022394582	.000021977

Then combine them in G:W=2:1 (Awoken) and G:W=1:2 (Displeased):

Awoken	Current Cookie
Last Cookie	Frenzy	Lucky	Clot	Ruin	E. Frenzy	Chain	C. Frenzy	Blab
Frenzy	0.0638090	0.6579225	0.0949378	0.0949378	0.0196663	0.0196663	0.0382750	0.0000372
Lucky	0.5629846	0.0827966	0.1279198	0.1279198	0.0235329	0.0347289	0.0400785	0.0000389
Clot	0.3190452	0.4469650	0.0189876	0.1279198	0.0235329	0.0321957	0.0313234	0.0000306
Ruin	0.3190452	0.4469650	0.0189876	0.1279198	0.0235329	0.0321957	0.0313234	0.0000306
E.Frenzy	0.3190452	0.4178495	0.0988044	0.0988044	0.0039333	0.0317477	0.0297865	0.0000291
Chain	0.3215784	0.4208307	0.0992524	0.0992524	0.0235329	0.0055762	0.0299479	0.0000292
C.Frenzy	0.3278003	0.4245416	0.0967413	0.0967413	0.0199330	0.0283091	0.0059040	0.0000294
Blab	0.3190452	0.4139830	0.0949378	0.0196663	0.0196663	0.0278811	0.0295199	0.0000288

Displeased	Current Cookie
Last Cookie	Frenzy	Lucky	Clot	Ruin	E. Frenzy	Chain	C. Frenzy	Blab
Frenzy	0.0319045	0.4713680	0.1898757	0.1898757	0.0393327	0.0472791	0.0303348	0.0000296
Lucky	0.2814923	0.0698796	0.2558396	0.2558396	0.0470658	0.0559082	0.0339418	0.0000330
Clot	0.1595226	0.4153622	0.0379751	0.2558396	0.0470658	0.0546416	0.0295642	0.0000289
Ruin	0.1595226	0.4153622	0.2558396	0.0379751	0.0470658	0.0546416	0.0295642	0.0000289
E.Frenzy	0.1595226	0.3571313	0.1976087	0.1976087	0.0078665	0.0537456	0.0264905	0.0000259
Chain	0.1607892	0.3592939	0.1985047	0.1985047	0.0470658	0.0092025	0.0266132	0.0000260
C.Frenzy	0.1639001	0.3573828	0.1934826	0.1934826	0.0398660	0.0466685	0.0051914	0.0000258
Blab	0.1595226	0.3493982	0.1898757	0.1898757	0.0393327	0.0460125	0.0259572	0.0000258

The final normalized table are presented below:

Awoken	Current
Last	Frenzy	Lucky	Clot	Ruin	E. Frenzy	Chain	C. Frenzy	Blab
Frenzy	2.090926%	21.55913%	3.110970%	3.110970%	0.644436%	0.996633%	1.254214%	0.001217%
Lucky	21.20543%	3.118624%	4.818239%	4.818239%	0.886392%	1.308101%	1.509599%	0.001465%
Clot	3.292897%	4.613170%	0.195973%	1.320273%	0.242885%	0.332295%	0.323291%	0.000315%
Ruin	3.292897%	4.613170%	1.320273%	0.195973%	0.242885%	0.332295%	0.323291%	0.000315%
E.Fren.	0.692677%	0.907191%	0.214514%	0.214514%	0.008539%	0.068927%	0.064669%	0.000063%
Chain	1.015420%	1.328821%	0.313401%	0.313401%	0.074308%	0.017608%	0.094564%	0.000092%
C.Fren.	1.177110%	1.524501%	0.347392%	0.347392%	0.071578%	0.101656%	0.021201%	0.000106%
Blab	0.001140%	0.001480%	0.000339%	0.000339%	0.000070%	0.000100%	0.000106%	0.000000%
Total	32.76850%	37.66609%	10.32110%	10.32110%	2.171094%	3.157614%	3.590935%	0.003574%

Displ.	Current
Last	Frenzy	Lucky	Clot	Ruin	E. Frenzy	Chain	C. Frenzy	Blab
Frenzy	0.559120%	8.260622%	3.327530%	3.327530%	0.689297%	0.828557%	0.531611%	0.000518%
Lucky	8.745448%	2.171032%	7.948467%	7.948467%	1.462247%	1.736964%	1.054508%	0.001026%
Clot	3.103346%	8.080440%	0.738767%	4.977094%	0.915615%	1.062995%	0.575141%	0.000562%
Ruin	3.103346%	8.080440%	4.977094%	0.738767%	0.915615%	1.062995%	0.575141%	0.000562%
E.Fren.	0.698358%	1.563449%	0.865091%	0.865091%	0.034438%	0.235287%	0.115970%	0.000113%
Chain	0.822308%	1.837500%	1.015193%	1.015193%	0.240704%	0.047063%	0.136105%	0.000133%
C.Fren.	0.492380%	1.073631%	0.581251%	0.581251%	0.119763%	0.140199%	0.015596%	0.000078%
Blab	0.000477%	0.001045%	0.000568%	0.000568%	0.000118%	0.000138%	0.000078%	0.000000%
Total	17.52478%	31.06816%	19.45396%	19.45396%	4.377797%	5.114199%	3.004149%	0.002992%

Sugar Lump

The method of picking sugar lump type is different from the above methods. The type determination procedure is as follows.

• Add Normal to a pool.
• If you have upgrades Sucralosia Inutilis, 15% chance to add Bifurcated to the pool, otherwise 10% chance to add Bifurcated to the pool.
• 0.3% chance to add Golden to the pool.
• 0%, 10%, 20% and 30% to add Meaty to the pool if the game is in Appeased, Awoken, Displeased and Angered phase of Grandmapocalypse respectively.
• Pick a random effect from the pool.

Unlike the previous examples, the random number 0<P<1, which decides whether the candidate should be added into the pool or not, is generated only once through the whole process.

Type	Probability
Normal	PN=1
Bifurcated	PB=0.1 or 0.15
Golden	PG=0.003
Meaty	PM=0.1×grandmas' anger level =0, 0.1, 0.2, 0.3

If P<P_i, then the i type will be added to the pool.

When P_h>=P_i>=P_j>=P_k, probability of selecting a h type: (P_h-P_i)+(P_i-P_j)/2+(P_j-P_k)/3+P_k/4, i type: (P_i-P_j)/2+(P_j-P_k)/3+P_k/4, j type: (P_j-P_k)/3+P_k/4 and k type: P_k/4.

Then we can calculated all the results.

Phase of Grandmapocalypse	Probability without Sucralosia Inutilis				Average number of lumps
	Normal	Bifurcated	Golden	Meaty
Appeased	0.9495	0.0495	0.001	0	1.02825
Awoken	0.933083	0.033083	0.00075	0.033083	1.01916
Displeased	0.883083	0.033083	0.00075	0.083083	1.01916
Angered	0.833083	0.033083	0.00075	0.133083	1.01916

Phase of Grandmapocalypse	Probability with Sucralosia Inutilis				Average number of lumps
	Normal	Bifurcated	Golden	Meaty
Appeased	0.9245	0.0745	0.001	0	1.0426125
Awoken	0.908083	0.058083	0.00075	0.033083	1.03311875
Displeased	0.87475	0.04975	0.00075	0.07475	1.02874375
Angered	0.82475	0.04975	0.00075	0.12475	1.02874375