Odds in Pokemon – by Xact
January 17, 2005
www.pojo.com/pokemon
Errata from Part 2
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
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 60card deck,
as I had hinted in Part 2. I know – the title above is corny. It’s a reference
to a YuGiOh! 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, biggerthanusual, 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=6014). That
means that, in roughly 2 out of 3 games, your maxedout 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:
XAct’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
oddsrelated 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!