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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 and Sugar Lump. 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.

Sugar LumpEdit

Take the type of Sugar Lump as an example. The type determination procedure is as follows.

  • Add Normal to a pool.
  • If you have upgrades Sucralosia Inutili, 10.5% chance to add Bifurcated to the pool, otherwise 10% chance to add Bifurcated to the pool.
  • 0.1% 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.

Hence there are four possible pools.

Pool Probability
Normal meaty Bifurcated Golden
X P1=1
X X or none P2=0.1×grandmas' anger level =0, 0.1, 0.2, 0.3
X X or none X P3=0.1 or 0.105
X X or none X X P4=0.001

When Ph>=Pi>=Pj>=Pk, probability of selecting a h type: (Ph-Pi)+(Pi-Pj)/2+(Pj-Pk)/3+Pk/4, i type: (Pi-Pj)/2+(Pj-Pk)/3+Pk/4, j type: (Pj-Pk)/3+Pk/4 and k type: Pk/4.

Then we can calculated all the results.

Phase of

Grandmapocalypse

Probability without Sucralosia Inutili
Normal Bifurcated Golden Meaty Expected value of sugar lumps
Appeased 0.94983 0.04983 0.00033 0 1.026083
Awoken 0.93325 0.03325 0.00025 0.03325 1.0175
Displeased 0.88325 0.03325 0.00025 0.08325 1.0175
Angered 0.83325 0.03325 0.00025 0.13325 1.0175
Phase of

Grandmapocalypse

Probability with Sucralosia Inutili
Normal Bifurcated Golden Meaty Expected value of sugar lumps
Appeased 0.94733 0.05233 0.00033 0 1.0286416
Awoken 0.93075 0.03575 0.00025 0.03325 1.01964375
Displeased 0.882416 0.034916 0.00025 0.082416 1.01920625
Angered 0.832416 0.034916 0.00025 0.132416 1.01920625

Force the Hand of FateEdit

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 Chain, 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 Chain=Storm=Blab Building Sugar
0.2973144554 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 Chain=Storm=Blab Building StormDrop Sugar
0.2527172871 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 cookieEdit

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 C CF B
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 C CF B
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%

Edit

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%

GrandmapocalypseEdit

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

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