Ness's Nest
with Jason Klaczynski
Opening Basic Math
December 6, 2006
During the next few weeks, I'll be responding to math
questions you guys may have regarding opening hands. (If
you're interested in submitting a question, read the
bottom.) This week, a friend asked me the odds of opening
with a certain Pokemon in his Muk/Weezing deck.
If
you're not a math person, you can skip right down to the
numbers. But if you'd like to learn how to do this
yourself, I'll walk you through it.
My
friend runs the following basic Pokemon in his Muk/Weezing
deck:
-4 Grimer
-2 Koffing
He wants to know the odds that he will be forced to open
with Koffing - that is, he'll get a Koffing in his opening
hand, but no Grimer.
The first
thing we have to do is determine the odds of drawing a
Koffing in our top seven cards. The simple way to figure
out the odds of drawing something are to first calculate the
odds of NOT drawing it. Then, we'll simply subtract our
answer from 1.
Step 1: The Odds of drawing Koffing
Since we play 60 cards, we know
the odds of avoiding a Koffing in our first card would be
58/60 (or 29/30). If we avoided the Koffing in our first
card, the odds of avoiding it in our second card would be
(57/59). We then have (56/58) for our third card, (55/57)
for our fourth, and so on, so that we get a total of
seven quotients. (One for each card in our hand.) In
order to get the probability of ALL of these things
happening, we take these seven quotients and multiply them
all together.
(58/60)*(57/59)*(56/58)*(55/57)*(54/56)*(53/55)*(52/54) =
.7785
This means the odds of avoiding the Koffing in our opening
hand are 77.85%. To determine the odds of getting the Koffing, we simply do 100% - 77.85%, or 1 - .7785.
1 - .7785 = .2215
So now we know there is a 22.15% chance we will draw
a Koffing in our opening hand. But that doesn't necessarily
mean we'll open with it, because we can draw one of our
other basics (4 Grimers) to save us from opening with
Koffing.
Step 2: The Odds of Drawing no Grimer with
our Koffing
Now, let's assume we did get a
Koffing in our opening hand. We would have six other cards
in our hand that could be a Grimer, and will save us from
opening with Koffing. We need to determine the chance that
we do NOT get a Grimer. When we have the probability of
drawing a Koffing, and the probability of not drawing
Grimer, we will be able to multiple these two results
together, and that will give us the final result. (For
now.)
The odds of avoiding a Grimer in our remaining 59 cards
(Remember, we have 59 cards left to work with, since we know
1 of them is a Koffing, and is in our hand) are 55/59. If
we avoid Grimer in our first card, the odds of avoiding it
in the second card are 54/58, 53/57 in the third card and so
on. We need a total of six quotients, since we have
six cards left to draw.
Odds of no Grimer in remaining 6 cards:
(55/59)*(54/58)*(53/57)*(52/56)*(51/55)*(50/54) = .6434
The odds
of not getting a Grimer with our Koffing are 64.34%.
Now, we
need to know the odds of both of these things happening.
Step 3: The Odds of drawing Koffing, and no
Grimer:
This is the one simple step in our process. If you want to
know the odds of one or more things all happening, all you
have to do is multiply the chance of them happening
individually, together. Since drawing a Koffing is .2215
and not drawing a Grimer in the remaining six is .6434, all
we have to do is multiply .2215 by .6434.
.2215 * .6434 = .1425
So the
odds of getting the lone Koffing are 14.25%.
While this may seem like the final step, there is one more
thing you have to compensate for, which will give us our
final answer.
Step 4: The Odds of a Mulligan
What we've done in the
previous three steps has given us the odds of drawing a
Koffing in our opening hand without a Grimer. However, what
we haven't calculated, is the actual odds of opening with
Koffing.
Why? Because our calculations don't take mulligans into
consideration. While a hand of seven grass energy
technically doesn't have a Koffing, it will be shuffled back
in and could become a lone Koffing. To take mulligans into
consideration, we first need to determine the probability of
us mulliganing.
We figure
this out using the same math we used in Step2 - the odds of
not drawing a basic in seven cards would be (54/60) for the
first card, times (53/59) for the second card, and so on.
Odds of mulligan:
(54/60)*(53/59)*(52/58)*(51/57)*(50/56)*(49/55)*(48/54) =
.4586
We will
mulligan 45.86% of the time.
Step 5: Compensating for Mulligans
Since Step 3 had not
compensated for mulligans, we know that we will get a lone
Koffing 14.25% of the time.
Since we mulligan 45.86% of the time, we obviously will
not mulligan 54.14% of the time. (1 - .4586).
This is
where things get a little tricky. Since our first step told
us we get Koffing 14.25% of the time, but assumed we didn't
mulligan, we can cross multiply to figure out the chance a
mulligan results in a lone Koffing.
In other
words, 54.14% of the time we don't mulligan, and we get a
Koffing 14.25% of the time.
So the
remaining 45.86%, we will get a lone Koffing in a
proportional amount.
(.5414/.1425) * (.4586/x)
x is
representing the odds a mulligan results in a lone Koffing.
Cross
multiplying this gives us .5414x = .0654
We then
divide .0654 by .5414 to isolate x.
x = .0654/.5414
x = .1208
So the
odds that we will mulligan, and then get a lone Koffing are
12.08%.
Step 6: The Final Step
Now all we have to do is add
our result from Step 3 with our result from Step 5.
The odds
we do not mulligan, and get a lone Koffing are .1425.
The odds we do mulligan, and get a lone Koffing are .1208.
All we
have to do is add these two results for our final answer.
.1425 + .1208 = .2633
So our final
answer lets us know, the odds that we will be forced to open
with a Koffing are 26.3%.
E-mail me
any questions you have related to opening with a certain
Pokemon, and or combination of Pokemon/Trainers/Energy. If
I like your question, I'll post it and do the math for you.
To make my calculation possible, please provide an entire
deck list.
Good
luck!
-Jason Klaczynski (ness@pojo.com) |