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 ga