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Hi TL,
Recently I tried doing a video about odds and probability for the card game I play, the Pokémon TCG. I was trying to figure out the probability that you would have certain cards in your opening hand of 7 cards, but it turned out to be more difficult than I anticipated.
Why is it so difficult? Well, in Pokémon, you must have a Basic Pokémon in your opening hand of 7 cards to start the game, otherwise you must reveal your hand to your opponent, shuffle it in, and draw 7 new cards. So, if you are trying to figure out the odds of starting with a certain card that is not a Basic Pokémon, you must account for the times that you will start with that card AND a Basic Pokémon.
Just to lay down some ground rules, in the Pokémon TCG you must have exactly 60 cards in your deck. There are three different card types: Pokémon (Basic or Evolution), Trainers (Item, Stadium, or Supporter), and Energy (Basic or Special). You are allowed to play up to 4 copies of any cards (excluding Basic Energy).
You can see the video where I attempt to work all of this out here (talk relevant to the topic starts at around 17:50). I'll post the things I'm confused about here to see if anyone can help me out.
So, let's say I play 4 copies of Pokémon Collector, which is not a Basic Pokémon. What are my odds of starting with at least one? To figure this out, I tried what made sense. All you have to do is find 1 - (odds of not drawing the card), and you draw 7 cards at the start. So, in this example, 4/60 cards are Pokémon Collector, and 56/60 are not. Therefore, the formula would be...
1 - (56/60)(55/59)(54/58)(53/57)(52/56)(51/55)(50/54) = 39.95%
However, we must account for the fact that one of these 7 cards MUST be a Basic Pokémon. What is the best way to approach this? I had trouble figuring this part out. Eventually I decided to assume the first card was a Basic Pokémon, and then you have 6 opportunities to draw the Pokémon Collector. So, the formula would be this instead.
1 - (55/59)(54/58)(53/57)(52/56)(51/55)(50/54) = 35.66%
Is this correct? I'm not sure if the formula for finding the results is accurate, but it seems to make sense from intuition. I just worry that it assumes that we do not draw it on the first card, which makes the calculation incorrect.
Once we get past this, I am curious to know what the odds are of drawing the Pokémon Collector from your first 7 cards PLUS the card you draw at the start of your first turn. Although that doesn't seem difficult, you have to account for another mechanic of the game, which is placing out prize cards. After you draw your opening hand (which contains a Basic Pokémon), you then place out 6 face down "prize cards" aside, which you cannot access. (When one of your opponent's Pokémon is knocked out, then you take a prize card; one of the ways to win is to take all 6 of your prize cards. There are certain Poké-Powers that allow you to manipulate your prizes, but we don't have to worry about those for this problem.)
After we figure out the odds of drawing the Pokémon Collector at the start of the game, we then have to figure out the odds of one of them being in the next 6 cards (prize cards). If we manage to figure that out, then we can calculate the odds of drawing one at the beginning of the turn. Does anyone have any idea on how to figure this out?
To clarify, we start the game out by drawing 7/60 cards (1 of which must be a Basic Pokémon), bringing the size of the deck down to 53. Then, we place 6/53 cards aside that we cannot access, bringing the size of the deck down to 47. Finally, we draw 1/47 cards at the start of the first turn. What are the odds of drawing a card that you play 4 of (in these examples, it was Pokémon Collector) in your opening 7 cards + 1 at the start of your first turn?
Finally, the last problem I had was trying to figure out how often you would start with just one of a certain Basic Pokémon. It is fairly easy to calculate how often you will draw the Basic Pokémon in your first 7 cards, but then you have to factor in that sometimes you will draw the Basic Pokémon AND another Basic Pokémon. The problem and a solution is outlined here, but I want to be sure that it is correct.
Thank you to anyone who can provide help! If you need me to clarify anything, please let me know.
-Pooka
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But if you don't have a basic pokemon on then you reshuffle. So really you can just completely ignore that card, right?
Looks to me that he second probability is correct.
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i would think that because you have to have one basic pokemon card, you could just say that you have a 6 card hand, and assume that the last card is a basic pokemon. this is becuase the hand would just be reshuffled and drawn again if you have none.
so the chances of getting at least one pokemon trainer card AND one basic pokemon are:
(4/60)+(4/59)+(4/58)+(4/57)+(4/56)+(4/55)=about 41-42% by my calcs
...right? I'm not good at probability either.
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I don't see how the prize cards matter. All you care about is the card on top of the deck. Does it matter if it's on the bottom of the deck or the prize pool? So shouldnt the probability of drawing it on the first turn just be 4/53?
I think I need more detail on the last problem.
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If you have Microsoft Excel, the function you want is 'HYPGEOMDIST(A,B,C,D)'. It's short for 'hypergeometric distribution', which is the distribution that describes the probability of drawing (without replacement) exactly A successes when drawing B cards from a deck which contains C successes and D total cards.
For example, if a 60-card deck has 4 copies of a certain card, and you draw 7 cards, what is the probability of getting exactly 1 of that certain card?
=HYPGEOMDIST(1,7,4,60)
=0.33628
So, there's a 33.628% chance of getting exactly one copy when drawing 7 cards. If you want to find the probability of drawing *at least* 1 copy, you must add the probabilities of drawing 1, 2, 3, and 4 copies together.
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Add conditional probability if you want to account for mulligan.
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DoubleReed: If you draw the Pokémon Collector but 0 Basic Pokémon, then you must shuffle the hand back in, and it is worthless. Therefore, it seems that basically we are drawing a 6 card hand (+ 1 Basic) at the beginning instead of a 7 card hand. After more research and checking math, I agree that the second calculation is correct.
For the last problem, it is fairly tricky to understand. I'll try to break it down into steps.
1) Draw 7 cards from the top of your deck. If you do not draw a Basic Pokémon in these 7 cards, reveal your hand to your opponent, shuffle it in, and draw 7 cards again from the top of your deck. Repeat until you get a hand with a Basic Pokémon. (We found the calculation for this part already, but now we are looking to find the odds of drawing the Pokémon Collector from the first 7 cards OR the card we draw at the beginning of the first turn.)
Cards remaining in deck: 53
2) Place the top 6 cards from your deck to the side. These are your prize cards, which you are not allowed to access. (The reason this is relevant is because these 6 cards could contain 1 or more Pokémon Collector, reducing the odds that we draw it from the deck. You are not allowed to draw from the prize cards.)
Cards remaining in deck: 47
3) At the start of your turn, draw a card.
What are the odds of drawing a Pokémon Collector (4/60 cards) from either the opening 7 cards OR the 1 card drawn at the beginning of your first turn?
ishboh: This was my initial question. How do you account for the fact that the Basic Pokémon could be drawn at any point in the first 7 cards? I think you would just assume that 1/7 cards is a Basic Pokémon, then you calculate the odds of drawing your Pokémon Collector (4/59 remaining cards) in the next 6 cards. I'm not sure, though!
lithiumdeuteride: Yes, someone else brought this to my attention, but I wasn't sure if it applied to my problem. The only problem with your calculation is that it does not account for the fact that you must draw at least one Basic Pokémon in the opening 7 cards. Then the calculation would become the following:
=HYPGEOMDIST(1,6,4,59)
When you sum up the totals, it comes out to around 35.66% to have at least one Pokémon Collector in your opening hand, which is what I have listed. The work can be found here.
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This is how I'd do it. I won't do the actual math as I'll surely screw it up and you seem to know your probability, but I'll outline the method
![[image loading]](http://i.imgur.com/V4M3Y.png)
Entire box (A+B+C+D) = total number of ways I can draw 7 cards
Big circle (A+B) = total number of ways I can draw basic poke
Small circle (B+C) = total number of ways I can draw prize card
B = total # of ways I can draw prize card and have basic poke
A+B+C = total # of ways I can draw either basic poke or prize card (including invalid draws)
= (total # of ways 7 cards can be drawn) - (number of ways you have no basic poke or prize card)
A+B, and B+C you know how to calculate
B (your desired draw) = (A+B) + (B+C) - (A+B+C) = number of ways you can draw your desired hand
B / (A+B+C+D) = probability of drawing your desired hand in 7 cards
B / (A+B) = probability of drawing your desired hand assuming valid draw (what you tried to calculate)
Accounting for mulligans... I dunno a simple way, but if you're good with doing it this way, you can redo with 6 cards, 5 cards etc.
edit: misread, I thought with the mulligan, you draw 1 less card
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On January 27 2012 05:35 Pooka wrote:
Just to lay down some ground rules, in the Pokémon TCG you must have exactly 60 cards in your deck. There are three different card types: Pokémon (Basic or Evolution), Trainers (Item, Stadium, or Supporter), and Energy (Basic or Special). You are allowed to play up to 4 copies of any cards (excluding Basic Energy).
Wizards of the Coast makes this game, yes? These rules are identical to M:TG and assuming energy works like mana aren't significantly different.
Find some old Dojo articles, they go over the probabilities of topdecking certain cards really well. It's been a while since I was into Magic but sites like Starcitygames.com, MTGnews.com, and MTGzone.com can probably help.
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Purind: I'll read what you wrote again, but I'm confused. Which problem is that supposed to be for? Also, when you say "prize card," are you referring to the actual term "prize card" for the Pokémon TCG, or do you just mean "the card we want to draw"? The only reason prize cards are relevant to the last problem is because the card we want might end up in those, lowering the odds of drawing it from the deck on the first turn of the game.
Offhand: Yes, the game was made by Wizards of the Coast originally. The rules are fairly similar to M:TG, but the win conditions are different. One of the win conditions is taking all 6 of your prize cards, which is what makes calculating probability so difficult with Pokémon. Essentially you are removing 6 cards from your deck at random which you cannot access during the game. I don't think anything like that comes up in Magic.
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Oh I see, so you're asking if the first eight cards you have is a pokemon collector.
Well, just factor in the 1-4/53 into the original computation. It doesn't matter about the prize cards. If it's "outside the deck" or its "not the first card on top of the deck" it is the same thing. Think of it this way: You draw seven cards. And if you don't have a pokemon collector, then you're asking if a pokemon collector is the 7th card in the remaining 53-card deck. Well, clearly it's the same probability if it's the 1st card in the remaining deck.
So just do the same computation but for an additional card. The probability that the first eight cards and the first seven + fourteenth card is the same.
Edit: I don't mean to confuse you further. But the probability would only be different IF you looked at all the cards in the prize pool, and then calculated the chance of a pokemon collector in the extra card. But in that case, you would have additional information, and it would just be a simple calculation, because it is not a calculation about the entire part, but just that one card.
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Prize card should be "Target" card as in the card you want along with basic poke. I don't know anything about Pokemon TCG aside from the fact that you have to attach energy to pokemon to use attacks or something... I understood it as "I want to draw basic poke and a copy of target card."
What I wrote was unnecessary though. Your method works as long as you can re-draw 7 cards if you have no basic poke in your hand. I thought that if you redraw, you had to draw 1 less so I had to include the cases where you draw an invalid hand
Your method of assuming 1 card is basic poke and taking 1 basic poke out of your 60 card deck works, since you're throwing away any possibility of not drawing 1 basic poke (and 7 card redraw allows you to throw those invalid hands away completely).
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There is a great article on it for the Wow tcg that should still apply with a 60 card deck,
http://www.wowcardfan.com/2012/01/14/the-opening-hand/
With 4 of a card in you deck and a seven card hand you have about a 40% chance to draw it. Goes over the math a bit too if you want to adjust numbers.
Hope that helps.
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Bookmarked. I took a probability course last semester and though I won't get to this anytime soon, it'd be fun to try in my spare time.
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EPT
