Pot of Duality

Haven’t really got much to post nowadays with the meta getting settled down in the middle of the format.

I just read Max’s post on how Upstart Goblin helps decks like Six Samurais to get their combo pieces faster. This reminded me of some calculations I did when I was testing out triple Pot of Dualities and triple Upstart Goblins in my Hero Beat deck a few months back.

During my free time, I actually did an entire Excel worksheet showing how much PoD and Upstart Goblin helped the deck in terms of consistency. However, my notebook has been down for quite some time and I didn’t bother to fix it yet, so I don’t have the file with me.

So, I will just be going through some minor calculations.

The deck I will be using as an example will be Blackwings as I do not wish to go crazy by repeating what I did with my Hero Beat deck (I calculated the probability of drawing Upstart through Upstart through Upstart through PoD, so the numbers were extremely small and almost negligible). Also, PoD’s once per turn effect makes it so much easier to calculate than Upstart Goblin.

Forgive me for any mistakes as I am doing this entire post with just my phone.

For starters, the deck plays triple PoD with a lone Black Whirlwind. I will be taking the starting hand as 6 cards.

Firstly, I will calculate the percentage of drawing Black Whirlwind in my starting hand without the help of other cards.

% of drawing Black Whirlwind in starting hand = (1/40)+(1/39)+(1/38)+(1/37)+(1/36)+(1/35) = 16.033%

Next, I will calculate the chance of drawing PoD in my starting hand.

% of drawing Pot of Duality in starting hand = (3/40)+(3/39)+(3/38)+(3/37)+(3/36)+(3/35) = 48.099%

Then, I will find the percentage of using PoD to get to Black Whirlwind.

% of PoD into Black Whirlwind = (1/34)+(1/33)+(1/32) = 9.0964% (This is why PoD is amazing, almost 10% to get a card which you only play a lone copy of)

Now, I will multiply the percentage of drawing PoD by the percentage of using PoD to get to Black Whirlwind.

% of drawing PoD and getting Black Whirlwind with it = 48.099%×9.0964% = 4.375%

And the total percentage of getting Black Whirlwind T1 is calculated by adding up the percentage of drawing Black Whirlwind naturally and the percentage of getting Black Whirlwind with PoD.

= 16.033%+4.375% = 20.408%

That means that you will open with Black Whirlwind at least every 5 games in theory. Although this hardly happens in real life as there are many other factors to probability, you can’t just add up the numbers to 100%. If your luck is extremely bad, you may never ever draw Black Whirlwind even if you play like a hundred games.

This shows how imba Pot of Duality actually is, it actually gets better and better the longer the game lasts as your deck is thinner.

I’m not exactly sure how all this probability stuff works, so if you guys spot any mistakes please feel free to correct me.

*Edit
It seems that my calculations were wrong, so I will fix them here using stattrek.com/Tables/Hypergeometric.aspx as introduced by Max.

Probability of getting Black Whirlwind T1 = 15%

Population Size (Deck Size) = 40
Sample Size (Starting Hand) = 6
Number of Successes in Population = 1
Number of Successes in Sample = 1

Probability of getting PoD T1 = 34.02%

Population Size (Deck Size) = 34
Sample Size (Starting Hand) = 6
Number of Successes in Population = 3
Number of Successes in Sample = 1

Probability of PoD into Black Whirlwind = 8.82%

Population Size (Deck Size) = 34
Sample Size (Cards revealed through PoD) = 3
Number of Successes in Population = 1
Number of Successes in Sample = 1

Probability of getting Black Whirlwind T1 = 18%

15%+34.02%*8.82 = 18%