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Yu Yu Hakusho
Harry Potter
Vs. System

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 ()

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