Odds in Pokemon – by X-act
January 17, 2005
www.pojo.com/pokemon

 

 

Errata from Part 2

 

            Ness recently messaged me on AIM concerning my last article, where he pointed out very slight errors in the calculation of the table. The point is that the first 7 cards you draw cannot be any 7 cards, but one of them must be a Basic Pokemon. That means that one of your Basic Pokemon in your deck will never end up as one of your prizes. Consequently, 59 out of the 60 cards can end up as your prizes, not all of the 60. And that means that my early suspicion that the 6 prize cards depend on the first 7 cards you draw is actually partly correct.

 

Anyhow, here’s the revised table from last time. As you can see it’s identical for practical purposes… only the percentages differ by about 0.4% at most.

 

Number of the same card in your deck

Probability of that card not being in your prizes

Probability of that card appearing once in your prizes

Probability of that card appearing twice in your prizes

Probability of that card appearing three times in your prizes

Probability of that card appearing four times in your prizes

1

9 out of 10 games (89.83%)

1 out of 10 games (10.17%)

 

 

 

2

8 out of 10 games (80.54%)

3 out of 16 games (18.59%)

1 out of 114 games (0.88%)

 

 

3

18 out of 25 games (72.06%)

1 out of 4 games (25.43%)

1 out of 41 games (2.45%)

1 out of 1625 games (0.06%)

 

4

16 out of 25 games (64.34%)

3 out of 10 games (30.88%)

1 out of 22 games (4.54%)

1 out of 429 games (0.23%)

1 out of 30342 games (0.003%)

 

Thanks to Ness for pointing out this slight mistake.

 

Part 3 – The Heart of the Cards

 

            In this third part of ‘Odds in Pokemon’, we shall look at the odds of getting a particular card depending on how many cards you have left in your 60-card deck, as I had hinted in Part 2. I know – the title above is corny. It’s a reference to a Yu-Gi-Oh! catchphrase, where ‘believing in the heart of the cards’ means believing that you will draw whatever you want at exactly the right time. Unfortunately, this doesn’t always happen in practice (in fact, it barely ever happens, to me that is)… but that’s why I’m writing this article; to see why it doesn’t always happen.

 

            Before starting, I wish to thank all those people who emailed me, telling me how much they like my articles and how useful they are finding them. I really appreciate your positive comments.

 

Starting the game

 

            When you start a game, you shuffle your deck and draw 7 cards, then (unless your opponent mulligans) you set aside 6 more cards as your prizes. That means that your deck has already 13 (7+6) less cards than what you started with, even before drawing your first card during the game. This is 47 cards (even less if your opponent mulliganed and you drew at least one card).

 

            Let me now present you with the following, bigger-than-usual, table, containing all the different odds of getting a card if you have 1, 2, 3 or 4 of any card remaining in your deck. Later on in this article, I will also present you my theorem related to this subject.

 

Cards left in your deck

Probability of getting a card if you have 1of it in your deck

Probability of getting a card if you have 2 of it in your deck

Probability of getting a card if you have 3 of it in your deck

Probability of getting a card if you have 4 of it in your deck

47

23.33%

41.53%

55.64%

66.54%

46

25.00%

44.07%

58.53%

69.45%

45

26.67%

46.55%

61.30%

72.16%

44

28.33%

48.98%

63.94%

74.69%

43

30.00%

51.36%

66.45%

77.05%

42

31.67%

53.67%

68.85%

79.23%

41

33.33%

55.93%

71.13%

81.26%

40

35.00%

58.14%

73.29%

83.13%

39

36.67%

60.28%

75.35%

84.86%

38

38.33%

62.37%

77.29%

86.46%

37

40.00%

64.41%

79.14%

87.92%

36

41.67%

66.38%

80.87%

89.26%

35

43.33%

68.31%

82.51%

90.49%

34

45.00%

70.17%

84.06%

91.61%

33

46.67%

71.98%

85.51%

92.63%

32

48.33%

73.73%

86.86%

93.55%

31

50.00%

75.42%

88.14%

94.38%

30

51.67%

77.06%

89.32%

95.13%

29

53.33%

78.64%

90.43%

95.80%

28

55.00%

80.17%

91.45%

96.40%

27

56.67%

81.64%

92.40%

96.93%

26

58.33%

83.05%

93.28%

97.41%

25

60.00%

84.41%

94.09%

97.82%

24

61.67%

85.71%

94.82%

98.18%

23

63.33%

86.95%

95.50%

98.50%

22

65.00%

88.14%

96.11%

98.77%

21

66.67%

89.27%

96.67%

99.01%

20

68.33%

90.34%

97.17%

99.21%

19

70.00%

91.36%

97.62%

99.37%

18

71.67%

92.32%

98.01%

99.51%

17

73.33%

93.22%

98.36%

99.63%

16

75.00%

94.07%

98.67%

99.72%

15

76.67%

94.86%

98.94%

99.79%

14

78.33%

95.59%

99.16%

99.85%

13

80.00%

96.27%

99.36%

99.90%

12

81.67%

96.89%

99.52%

99.93%

11

83.33%

97.46%

99.65%

99.96%

10

85.00%

97.97%

99.75%

99.97%

9

86.67%

98.42%

99.84%

99.986%

8

88.33%

98.81%

99.90%

99.993%

7

90.00%

99.15%

99.94%

99.997%

6

91.67%

99.44%

99.97%

99.9990%

5

93.33%

99.66%

99.988%

99.9998%

4

95.00%

99.83%

99.997%

100.00%

3

96.67%

99.94%

100.00%

 

2

98.33%

100.00%

 

 

1

100.00%

 

 

 

 

            Suppose you have, say, 4 Steven’s Advice in your deck. Then the probability that you draw a Steven’s Advice among the first 14 cards of your deck (the opening hand, 6 prizes, and top card of the deck) is 66.54% (since you have 47 cards left in your deck before you draw your first card off your deck, making them 46=60-14). That means that, in roughly 2 out of 3 games, your maxed-out card is either in your opening hand, or is in your prizes, or is the first card you draw when you start the game. Or say you have 3 Gust of Wind in your deck. Then the probability of having a Gust of Wind among the first 30 cards of your deck is 88.14%, or roughly in 8 games out of 9. This will have some implications later on.

 

Comments on the above table

 

            You might think that some of these odds are too high, especially for when you have 3 or 4 of the same card. Indeed they are quite high, and that means that, under normal circumstances, you would have already drawn at least one of them even before you draw your very first card off your deck after drawing your opening hand and placing your 6 prize cards (they might be in your prizes, but, for our intents and purposes, you have drawn them). In fact, the probabilities that you have already drawn a card you have 3 or 4 of in your deck are always greater than 50%, no matter how many cards you have left in your deck. You may try a little experiment if you want. Put 4 of a particular card in a deck of 60 cards, then shuffle the deck and draw 14 cards, and repeat for as much as you like. You’ll find that in roughly 2 out of 3 trials, that card you put 4 of is among the 14 cards you drew, because the probability is 66.54%. (You can use Apprentice to speed things up.)

 

            A very important point in the Pokemon TCG is that you don’t know your real odds of drawing a particular card during the game unless you know which cards are in your 6 concealed prizes. This makes the Pokemon TCG differ from most other TCGs; you have 6 inaccessible cards of your deck, that you can ONLY access once you manage to knock out one of your opponent’s Pokemon (barring exceptions like using Tangela from FRLG). And you cannot know what cards you have in your prizes until you draw them (without using Here Comes Team Rocket or Team Rocket Rattata, or other similar cards).

 

            Wait a bit. You CAN know what prize cards you have! You can look at your deck and, from the missing cards you have in your deck, conclude which cards you have as your prizes. And how do you look at your deck? You can by playing a Trainer Card that allows you to. I’ll list a few here:

 

Computer Search

Oracle

Celio’s Network

Lanette’s Net Search

Wally’s Training

Archie

Dual Ball (75% of the time)

Great Ball

Lady Outing

Pokemon Fan Club

Pokemon Trader

Professor Elm’s Training Method

Swoop! Teleporter

Surprise! Time Machine

Rocket’s Pokeball

Team Magma/Aqua Ball

Team Magma/Aqua Conspirator

 

            And the list goes on and on. Most of these Trainer cards are commonly played in good Unlimited and Modified decks. There are also Pokemon that allow you to look at your deck; Dunsparce from Sandstorm is the most common of them. So, as soon as you play one of these Trainer cards for the first time, grab the opportunity to look over your deck carefully, to see (roughly) what cards are in your prizes. I put the word ‘roughly’ in there because, under tournament conditions, you usually won’t have the time to do this thoroughly. However, it’s a good idea to have a rough idea as to what prize cards you have, without delaying your game too much.

 

            Now that you roughly know which cards are in your prizes, you can know more about the odds of drawing specific cards. Suppose you see that you have roughly half your deck remaining on the table. Say you had already drawn one of your 4 Steven’s Advice and used it, and you haven’t drawn another one. According to the table above, if you have 30 cards left in your deck, you have very high odds of having drawn a card you have still 3 of among your first 30 cards; they are almost 90% (89.32% is more accurate). However, if you know that you have one of them in your prizes, then you know that you have already drawn it.

 

            We can also use the above table to have an idea of which cards you have in your prizes without even looking at your deck. As you might have noticed when you were looking at the table, I made all odds over 90% in boldface. That means that those are very high odds of drawing the card you want. If you have only one of a card in your deck, these odds start after you have 7 or less cards in your deck. If you have 2 of a card in your deck, these odds start at 20 or less cards. If you have 3 of a card in your deck, these odds start at 29 or less cards, and if you have 4 of a card in your deck, these odds start at 35 or less cards.

 

            Suppose, again, that you have 4 Steven’s Advice in your deck, and that, after having 35 cards remaining in your deck, you still haven’t drawn one. You can conclude that it is almost certain that one of them is in your prizes. You can conclude the same thing if you have 3 of the same card, if you have 29 cards remaining in your deck, and so on for when you have 2 or 1 of the same card. Here is, then:

 

X-Act’s Theorem in Pokemon TCG Card Drawing

 

If you have 3 or 4 of the same card in your deck, and you have still not drawn one of them after drawing half of your deck, then there is a very high probability that at least one of those cards is among your Prize cards.

 

            I’m not saying anything spectacular in this theorem, nor is it too enlightening. But people tend to forget the obvious while playing, especially if they’re under tournament conditions. Instead of becoming frustrated about not being able to get your maxed out card after drawing half your deck, remember this theorem: you have probably at least one of them in your Prizes. If you happen not to have any in your prizes, then yes, you have every reason to be frustrated, since you are very unlucky to have all four of the same card at the bottom half of your deck (even after shuffling repeatedly).

 

            What’s cool about this theorem is the fact that it can be applied at a relatively early stage of the game. Remember that you start your game with 47 cards in your deck, so after playing a couple Professor Oak or Steven’s Advice, you have already almost half your deck out, so you can start thinking about which cards you haven’t yet seen out of those you have 3 or 4 of. Chances are you have a few of them in your Prizes.

 

            (If you have 2 of the same card in your deck, my theorem can be applied after drawing two thirds of the deck.)

 

Conclusion

 

            The next article will concern Energy cards. How many should I play in my deck? Why am I not getting Energy at the right time? Why am I getting too many Energy cards in my starting hand? We’ll answer these questions in next week’s article, and more.

 

            If you have any questions (like from where I got those values, etc.), or you want to know the odds of anything in particular, feel free to email me at xactcreations@yahoo.com. You can also AIM me (my nickname is xactxx). I might not be able to answer you immediately because of my work commitments, but I promise that I’ll answer as soon as I can. Also, if you ask me an odds-related question, it might not be easy for me to have an answer in 2 minutes… some questions require a pretty hefty amount of time for me to calculate. Don’t let this discourage you from asking me questions, though… let me get your emails!