Streaming Live Games on Pokémon TCG Online (Offline)
- Thread starter Pooka
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Notes:
Pokémon in starting hand: this chart was unreadable, but everything sounded right.
Those numbers are all based of off a 7 card hand. If you're not mulliganing, you should be looking at 6 cards not 7, as 1 card is a Pokémon. Of course, you then draw a card, but this card comes from a differently apportioned pool based on the results of what's in the opening hand, so that'll get pretty unwieldy. (You do note this in the following Collector example).
However your math on Collector is wrong. Hypergeometric distribution is the function you want to use when looking for non-replacement based success from a fixed population. You can check my example here google docs link. Each of the first 4 entries are the odds of getting exactly 1,2,3, and 4 copies of collector w/ 4 in your deck out of 6 cards. The lower one is the sum of those possibilities (since the others are exacts, this is correct to sum here). So the odds of getting a Collector in your example is actually an even more grim ~35%.
It looks like the rest of your math suffers from the lack of Hypergeometric distribution.
Pooka
Master Trainer
Unfortunately, I realized that the math for opening with Collector was incorrect after I had finished. I accidentally had it set to draw a seventh card when I had meant to draw six (assuming one was a Basic). After doing the math again, the number I got was 35.66%. Thanks for the help! I'm actually not very good with statistics, but people requested that I do a video about it, so I tried my best to act smart for 30 minutes. I hadn't even considered Hypergeometric distribution.
Yes, 35.66% is exactly correct. I reflexively had put 60 instead of 59 as the population size, but if you're assuming a non-mulligan and only using 6 cards for your sample, your population should be 59.
Also I don't have time to muscle through it right now, but i think you can ignore mulligans in the "solo Rotom" situation, since you reboot the universe if you get a mulligan. I think it should just be HYP(1,7,1,60) * HYP (0,6,4,59) (odds of both 1 Rotom and 0 Durant) ~ 7.5%.
I've updated that Google Doc w/ the fix to the first one and the math for the 2nd one.
Glumanda
Rulings Compendium Translator
I think you cannot ignore the Mulligans in the Solo-Rotom scenario.
Principally, you would have three cases:
a) Solo-Rotom = 7.5%
b) At least one Durant = x%
c) Mulligan = y% (this percentage should draw 0 energies out of 5 in the deck)
a) + b) + c) = 100%
You can have Solo-Rotom
1) In your first starting hand: Probability = 7.5%
2) After one mulligan in the second starting hand: Probability = y% * 7.5%
3) After two mulligans in the third starting hand: Probability = y% * y% * 7.5%
and so on.
So you would have to solve an infinite geometric row which should converge to some value. I must say that I do not know from my university time how to do this...
The result would be: 7.5% * sum of y% to the power of 0 to infinity (7.5% * (y%^0 + y%^1 + y%^2 + y%^3 + ...)).
Principally, you would have three cases:
a) Solo-Rotom = 7.5%
b) At least one Durant = x%
c) Mulligan = y% (this percentage should draw 0 energies out of 5 in the deck)
a) + b) + c) = 100%
You can have Solo-Rotom
1) In your first starting hand: Probability = 7.5%
2) After one mulligan in the second starting hand: Probability = y% * 7.5%
3) After two mulligans in the third starting hand: Probability = y% * y% * 7.5%
and so on.
So you would have to solve an infinite geometric row which should converge to some value. I must say that I do not know from my university time how to do this...
The result would be: 7.5% * sum of y% to the power of 0 to infinity (7.5% * (y%^0 + y%^1 + y%^2 + y%^3 + ...)).
Ah, yes. that should be Σz=0=>∞ y%^z * 7.5. Let me ask my friend Wolfram Alpha what he thinks about that.
(i updated my Google Doc w/ HYP (0,7,5,60) the "5 basic mulligan" which is ~52.5%
Wolfram Link
So that gives us around 15.7%?
I think the biggest lesson we have today is: TCG math is hard.
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Pooka
Master Trainer
Yeah, I was going to try to show the odds of drawing a Pokémon Collector in your opening hand OR the card you draw at the start of your first turn, but that opened up a whole new can of worms since 1-4 Collector could end up in your prizes before you draw that card... Didn't want to risk looking dumber than I already would.
sdrawkcab
Forum Moderator
Just like in poker where you don't take your opponents hole cards into consideration when calculating pot-odds you wouldn't take prize cards into consideration since they're all unknown variables.
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