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Topic: Deck Shuffling

Otaku

Member # 42359

posted July 16, 2002 02:16 PM

OKay, I have likely to made an idiot out of myself for posting this. Deck shuffling is a matter that I feel is worth looking at, and I will explain why. First, at my league, a lot of the youngr kids forget to shuffle a lot. Most think that letting one's opponent shuffle whenever they do something to their deck (Play an Oak, the Boss's Way, etc.) is some annoying house rule to frustrate them. Even those that do shuffle don't realize that they should use the 1 minute time-limit because a bad shuffle can cost you the game (technically, it could also win it... but it seems more than likely the cards that need to be together won't be). So I was wondering if anyone has a similar problem with youner players and how they approached it. Second, I have a special way of shuffling. So far, one and only one person has felt this was cheating, and that was someone who read how I did it. Those who have seen it have no problem. I will write out how I do it. Please read it carefully and let me know what you think (because I really don't want to find out at a major tourney):

First I organize my cards according to Pokemon, Trainer, and Energy. Then I further organize them in those groups. Next I combine thos three piles into one. Now I deal out 10 piles of 6. At this point, each pile has fairly equal amounts of each type of card). Now, I combine all 10 piles into one, and give my deck a thorough shuffling. I almost always do this in front of my opponent (the ones who I don't have already okayed it). Most people have no problem. The more you shuffle, the more I would think the cards would be evenly distributed. So all this does is allow a lot of shuffling in less time. But this is apparently similar to an magic practice using lands. So is this cheating? If it is... [Embarrassed]

The other reason I check is that an even distrubution helps reduce "card droughts" (when a certain type of card doesn't show up for a while). This helps reduce FTKOs, among other things. FTKOs are part of another topic, and as such made me think this might actually need discussed.

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From: Iowa | Registered: Sep 2001 | IP: Logged

tick

Member # 65651

posted July 16, 2002 03:49 PM

I believe that you have answered your own question in your post.

quote:
Originally posted by Otaku:
The other reason I check is that an even distrubution helps reduce "card droughts" (when a certain type of card doesn't show up for a while).

I'll infer from this comment that the reason you shuffle using this method is to attempt to reduce "card drought". This is basically the same thing when a Magic player "mana weaves" his/her deck. The problem is that you are attempting to create an "even distribution", which is not the same as shuffling.

Rule 21 of the UTR defines "shuffling" as making a deck "sufficiently randomized". The flaw of your reasoning is that you believe "sufficiently randomized" should create a evenly distributed deck. It does not. There have been many articles written about the science of "shuffling", I will not repeat them here.

While the technique that you use may not violate any tournament rules, provided that you shuffled the deck sufficiently afterwards, your intent may be cause for scrutiny.

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From: The City | Registered: Feb 2002 | IP: Logged

GreatFox

Member # 77642

posted July 16, 2002 09:23 PM

I kind of do the same thing.... But I only do that once in a while.... But never durring a game.

Before hand, like when I first build the deck and at least once a week, I too seperate the cards into Pokémon types (further arragned into evolution chains), Energy Types, and Trainers.

So say I have an Electric/Fighting deck. I would arragnge piles of cards like follows:

1. Electric Pokémon (Arranged in Evolution Cahin)
2. Electric Energy
3. Fighting Pokémon (Arranged in Evolution Cahin)
4. Fighting Energy
5. Colorless Energies (if aplicable)
6. Trainers

I then stack them on each other by taking one card at a time from each pile. Then I divide them again into 6 piles by placeing one card face down at a time in the following order:
1,2,3,4,5,6
2,3,4,5,6,1
3,4,5,6,1,2
4,5,6,1,2,3
5,6,1,2,3,4
6,1,2,3,4,5
1,3,4,6,2,5
6,4,3,1,5,2
2,5,3,1,6,4
1,2,3,4,5,6

And finaly stack these piles on top of each other in order (1,2,3,4,5,6) and a quick suffle.

I beleive this method compleatly randomizes the cards at the end while still keeping the cards even out through the Deck. With a second suffle before a game, the cards are no longer compleatly random

It should be OK to do any kind of suffling before games... but durring... you shouldn't... it should just be quick 30 seconds suffles.

.

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From: Los Angeles, California | Registered: Apr 2002 | IP: Logged

Big Daddy Snorlax

Member # 11

posted July 16, 2002 10:10 PM

Sorting the careds into piles and shuffling the piles together is something I always do. Six piles will come out even, with 10 cards in each pile, so it is also a double check to make sure that you haven't lost a card during the last game.

However, sorting the cards by type is something I would avoid. Whether it really does anything to manipulate your deck or not is debatable, but it could certainly look like you are stacking the deck. Even if you pile and shuffle afterwards, your opponent might very well think you stacked your deck, so I would avoid that step. Why give your opponent something to complain about?

BDS

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From: Oregon | Registered: Feb 2001 | IP: Logged

Full_Heal_Energy

Member # 10225

posted July 17, 2002 12:56 AM

quote:

I beleive this method compleatly randomizes the cards at the end while still keeping the cards even out through the Deck.

But those two concepts are mutually exclusive - if the cards are completely randomized, there SHOULD be some clusters of card types at some places in the deck. Having the cards be so regularly spaced is UNRANDOM.

quote:
With a second suffle before a game, the cards are no longer compleatly random

They are NEVER completely random, but the more you shuffle them, the better they'll be.

From: palo alto, california | Registered: Mar 2001 | IP: Logged

Otaku

Member # 42359

posted July 17, 2002 06:53 AM

Thank you for responding, and doing it politely: politeness is important when correcting a a mistake that is not life hreatening. I felt that one person who called me on this was insulting (note: they [probably] weren't, I was just being overly sensitive), and I nearly ignored the topic. I will cease pre-sortong since it is card-weaving (for a lack of a better term). I will likely still do the ten pile sort as this does "shuffle" better than conventional methods given my lack of manual dexterity, but will no longer "organize" the cards before. As I said, though to many this seems to be obvious cheating, not only have few few people minded, but our local tourney judges (who are not professors) would even give me time by to do it. Oh, and I only did it between games. A point to be further discussed is deck randomization, and how much shuffling actually randomizes. But now I would appreciate it if some of you could tell me how you shuffle, so I may properly instruct others. Thank you.

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From: Iowa | Registered: Sep 2001 | IP: Logged

CJ-Mich

Member # 835

posted July 17, 2002 07:13 AM

I still see many questionable shufflers out there even at Professor only events. Those that barely shuffle all, look at their cards while shuffling (the faces), etc, that I have made it a practice to almost always shuffle my opponents deck before a game.

So, in your case, I would have no objection to the way you randomized your deck. I would then shuffle your deck for you, and there would be no room for any complaints.

If we all got into the habbit of shuffling our opponents deck, at least at the start of the game, this wouldn't be near the issue that it is. I used to be worried that my opponent would think that I distrust him or her, but I have learned that they will have to understand it's not always a matter of mistrust, just a proceedure I go through to keep any thoughts of an un-sufficiently shuffled deck from keeping me from concentrating on the game at hand.

Sometimes players just don't shuffle good enough for me. Not that they are trying to cheat, it's just that maybe they don't appreciate the value of a well shuffled deck. A lot of the time I am probably doing them a favor.

And then when it comes to the younger players, you have to shuffle for them, because they just simply can't.

From: Michigan | Registered: Feb 2001 | IP: Logged

Pokidad

Member # 135

posted July 17, 2002 09:19 AM

CJ-Mich really gets to the point of this discussion.

We have seen and used several methods to shuffle decks. The most important one is to shuffle your opponent's deck if you feel that the way they are suffling looks [Dubious]

We have seen stack shuffling (prearranging the deck) a big [NoNoNo]

We always will shuffle these decks and them let our opponent cut their deck before beggining the game.

If we notice a player shuffling a deck during game play where they are looking at the cards as they shuffle, we will always cut the deck after their shuffle and then politely tell them that they are not allowed to see their cards as they shuffle. If they continue, call a judge over and explain the situation, politely, and asked them to observe the behavior. Usually corrects the situation real fast.

Separation of cards by type, then distributing in piles of 6 or 10, if fine, provided the deck is further shuffle. A lot of players will do this to try to obtain a more random distributions of cards (trainers, Pokemon, energies). This is really a good thing to do especially with some decks. Again, significant shuffling is necessary so that there is not an appearance of stacking the deck (prearranging cards).

As CJ_Mich indicated, you have the right to shuffle your opponent's deck (within reason) and be nice to their deck (don't try to look at their deck yourself to determine what your opponent is playing or bend their cards). You can even ask a Judge to assist you in this area or have the Judge observe your opponent in their shuffling of the deck. Most players have been very supportive in allowing us to shuffle their decks. As a matter of principle, we have always offer our opponents to shuffle our decks and at least cut our decks prior to any tournament, be it DCI or not. To us, proper manners in playing a card game.

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From: Falls Church, VA USA | Registered: Feb 2001 | IP: Logged

sentimental_blastoise

Member # 70533

posted July 17, 2002 09:28 AM

Just my 2 pence :-)
I believe your shuffling method is perfectly acceptable. I use a very similar method with low basic decks to decrease the chance of FTKOS, making sure I have no two basics next to each other before a blind shuffle. As to whether this is interpreted as stacking/marking, it's all your opponent's interpretation - I can see that performing your technique during a game might cause an eyebrow to be raised! Offering deck cuts usually solves it for me.

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From: United Kingdom | Registered: Mar 2002 | IP: Logged

Otaku

Member # 42359

posted July 17, 2002 11:06 AM

These latest responses bring up another point I wanted to focus on (and distract from my mistake [Wink]

): does any Professor not regularly cut or shuffle their opponents deck at the appropriate times (basically anytime your opponent has shuffled their own deck)? [Dubious]

I know many younger players don't, and I may not when time is rushed in an informal match, but otherwise it is a must. As I sometimes have to explain to younger players, its not that I don't trust them, but rather that because of the handful of people I have played who have shuffled poorly and got their card combos still intact from their discard pile, I must. I also mention that some try to cheat if I am not playing them (so as not to imply that they are cheating), but many of the younger children are as sensitive as myself.

Finally, it was mentioned that the purpose of shuffling is to sufficiently randomize their decka: who here actually has that in mind when they shuffle? That is to say, nearly every player I know shffles "their way" to try to get a good hand. In most cases, shuffling the maximum allotted time hoping for a-you guessed it-even distribution of cards. Most assume that if you shuffle a deck thoroughly, the odds are better that a card would be less likely to be around others like it than others unlike it, unless that card's "type" consists of over hald the deck. I have a similar question for cutting/shuffling/ignoring your opponents deck: while obviously less invasive then my shuffle (which I only used on other people decks who asked me to), don't most people try to cut in a manner that inconveniences their opponent. It is not very likely, but still most try.

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From: Iowa | Registered: Sep 2001 | IP: Logged

Full_Heal_Energy

Member # 10225

posted July 20, 2002 12:24 AM

quote:
Finally, it was mentioned that the purpose of shuffling is to sufficiently randomize their decka: who here actually has that in mind when they shuffle? That is to say, nearly every player I know shffles "their way" to try to get a good hand.

This actually surprises me -- I've always assumed the point of shuffling is to make your deck effectively random. I'm not trying to maximize my chances of any particular hand -- quite the contrary.

Now the word "effectively" may need some clarification. In fact it takes a lot more shuffling than almost anyone ever does to really come close to randomizing a deck in a meaningful way.

(A nice trick with a deck of playing cards: start with the cards sorted in order, then riffle-shuffle them three times, then, without looking at the cards, swap the position of two cards in the deck. By examining the deck you can almost always figure out which two cards were swapped. That's not a well-randomized deck!)

Still, if I need to shuffle my deck in the middle of the game -- say I do a computer search and pull out a card -- I'm willing to believe my deck is more or less random already, I just need to shuffle once or twice so I won't remember what card is where. They may not be truly randomized, but they are effectively randomized for the purpose of the game.

quote:

Most assume that if you shuffle a deck thoroughly, the odds are better that a card would be less likely to be around others like it than others unlike it...

Well, this is probably true IF you began with the deck sorted -- all similar cards next to each other. The next few shuffles will tend to separate them. But, on the other hand, if you begin with a deck in which you've artificially separated like cards, more shuffles will tend to bring them together somewhat. And in each case, the more shuffles, the closer to a really random distribution.

There's a good, somewhat relevant, story about a bridge (the card game) tournament that switched, some time in the 70s or 80s, to using computer generated bridge hands -- and the players were SURE something was wrong with the randomization process the hands seemed WAY more lopsided than they used to be -- players would have voids (no cards in a suit) and singletons (only one cards in a suit) far more often than they used to. But the fault was not in the computer randomization, but how the players previously used to shuffle the cards after every game. They'd gather up the tricks from the last hand -- already largely sorted into four cards of the same suit -- and shuffle once or twice. But this meant that far too
often there would be several groups of four cards of the same suit left together in the deck, guaranteeing that each player would get at least one. LESS shuffling led to MORE evenly spaced hands -- truly random decks would tend to give each player more clusters of single suits.

quote:

don't most people try to cut in a manner that inconveniences their opponent. It is not very likely, but still most try.

I'm not sure I understand what this means -- how can you cut in a manner that inconveniences your opponent? If he or she has truly shuffled the cards, then the cut is superfluous; if you feel he or she has shuffled in a way to bring desired cards near the top, then a cut is essential. But, without seeing the deck, how can you choose the best (or worst, from your opponent's point of view) cut?

[ July 20, 2002, 12:25 AM: Message edited by: Full_Heal_Energy ]

From: palo alto, california | Registered: Mar 2001 | IP: Logged

Otaku

Member # 42359

posted July 20, 2002 06:04 AM

Full_Heal_Energy had some questions as to what I was talking about with regards to comments on shuffling (which is good, since that is the topic of this thread). First, on how I said that most people I know shuffle thoroughly in the hope that the cardss, while random, will still be more or less evenly spread out. And the ones I know do. Do they actively say "I am shuffling to try for what is a random distribution of cards in the sense that the exact order is not known to anyone but am also hoping that the distribution will be fairly even". Rather, its comments like "I didn't shuffle enough. Look, no basic", indicating that had they thoroughly shuffled, they think that the cards would be more or less eveny distibuted, and they would have had a basic.

I assume Full_Heal_Energy to be a good exception to the norm (or else all of us Iowans are pure evil [Devilish]

). Part of the difference in expectations in shuffling comes from another point FHE disagreed with me on: I say that, even if the cards are already fairly evenly distributed throughout the deck, further shuffling will likely not produe a lot of card clusters, and those that do are even less likely to remain with further shuffling. Again, I base this on the fact that most decks in Neon Modified (they tend to be close to 20-20-20 splits on card types) have roughly even card types. This means that when I go to shuffle, there only a 1 in 3 chance that the card it ends up by on one side is of the same type, though it is not impossible for it to end up by a similar card. The odds are only 4 out of 9 if I calculated it correctly that it will end up between two different type of cards, but that is acceptable, becasue when you go to shuffle each time, the odds don't vary up or down of changing that. When we get to card subtpes (Basic or evo Pokemon, Trainer Types, Basic or Special Enrgy, etc.), the odds start to get lower, unless you have a disproportianate amount of one type, like Basic energy usually is. But other than the Basic Energy, it the odds keep dropping that any card, when randomely removed from any place in the deck and re-inserted randomly would be around its type when less than 1 in 6 cards of of its type. Finally, when you get to specific cards other than basic energy), the odds drop even more. Is it impossible for you to get threee bills in a row? No. BUt if you have three bills in a deck and randomly draw a card that happens to be a Bill out from said deck, then re-insert it totally randomly (thus allowing for it to be placed back in the same spot even), then there is only a 1-in-20 chance of it ending up by another Bill. I know this is rather mess, but hopefully after a few times through it everyone can undersdtand what I am saying. If not, can someonelse try and state it more eloquently? Thanks [Smile]

As for the trying to cut in a manner that inconveniences you opponent, most of the time, like I meant to say, it just means that you cut and hope they don't draw say the GoW they need to break your Mean Look. But if you watch carefuly, sometimes you can keep track of what they shuffled in, which even if it is illegal (can't see how you'd police it), is still a good reeason to shuffle thoroughly. Also, as many poeple know, the Foil cards tend to curve over time. This can be apparent, even with backs to those who really know what to look for. I keep my under other cards to help keep them flat, so that I amnot accused of marked cards, and my opponent can't use it against me. After all, several occaisions spring to mind when I don't need a Holo card, like when I have none of the lower stage cards left and its the end of a game.

[ July 20, 2002, 06:05 AM: Message edited by: Otaku ]

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From: Iowa | Registered: Sep 2001 | IP: Logged

Crobat1

Member # 85086

posted July 20, 2002 06:35 AM

This entire conversation has been quite interesting. I realize that many players do all kinds of "randomizing" in the interest of adequately shuffling their deck and unwittingly reduce the chance (inappropriately) of the deck being too "clustered". When my opponent asks me if I would like to cut at the beginning of a game, I have often been tempted to shuffle instead. Why? Being cutting doesn't accomplish what schuffling does. Generally, however, my wish to remain polite overrides my wish to insure that their deck has been randomized.

If you are playing 4 Crobats (my favorite), there is approxmately a 20% chance that 2 of them will be in the deck next to each other in a randomly sorted deck (details provided on request).

If you are playing 4 each of 8 different important cards, approximately 84% of the time, at least one of these importnat cards will be next to one of its copies.

My suggestion for those who care and who wish to test how randomly they are mixing the cards: test it out against the numbers. If your randomly shuffled decks do not offer 16% of touching twins, they are probably not randomized.

I hope this is helpful.

I truly believe that everyone involved in this conversation has the goal of fairness. There may be honest differences about how best to accomplish this goal.

By the way, I have encouraged my 10 year old, when he deals himself a Mulligan to put the original 7 cards into different parts of the deck (i.e. not next to each other). By my own logic, this too has a slight undermining of randomizization. It's a tough issue.

The only truly fair way of shuffling would be to each and every time, shuffle the deck many times to override the impact of card placement. And for younger kids who are not great shufflers, this means many, many times.

We do have to keep the game fun.

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From: Binghamton, NY | Registered: May 2002 | IP: Logged

ebArtemis410

Member # 380

posted July 20, 2002 05:10 PM

The piling into each type of card is good to do outside of battle, pre-game, that would kind of be like stacking, making sure you get a solid weave.

You could do that between games, and then pregame, do some riffles, then pile, riffles, then present your deck to your opponent, but piling into types I think is indeed manipulating the draw.

Like someone said, random doesn't mean perfectly distributed.

On a side note, I have a gripe about shuffling other people's decks. Today at a tourney, I present my deck for shuffling, with Japanese SUPER Silver sleeves. I use 60 for my Constructed deck and 40 for Limited, and I try to keep them in very good condition as I can't buy them here in town.

Well, the kid does the most BRUTAL shuffle I've ever seen in my life, and I'm like, about to cringe in my chair. REMEMBER! When you shuffle, they're not your cards, so be gentle!

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From: Raleigh, NC | Profession: Lamer | Registered: Feb 2001 | IP: Logged

Full_Heal_Energy

Member # 10225

posted July 20, 2002 11:33 PM

OK, Otaku, I think we're about as close to agreement as we're likely to get (or, for that matter, as we need to be). If by "try[ing] to cut in a manner that inconveniences their opponent", you meant just hoping your cut leaves them with bad cards, then I suppose I agree. I don't WANT my opponent to have a lucky draw, just like I want them to flip lots of tails -- but if I am capable of making conscious actions to increase the chances of either, something is wrong.

I do want to make a comment on something else you mentioned:

quote:

Rather, its comments like "I didn't shuffle enough. Look, no basic", indicating that had they thoroughly shuffled, they think that the cards would be more or less eveny distibuted, and they would have had a basic.

I think people often underestimate the probabilities of not getting a basic.

Here's a table -- the nth row shows the probabilities, when drawing 7 cards from a (well-shuffled) 60 card deck containing n basic pokemon, of drawing ZERO (first column), ONE (second column), or MORE-THAN-ONE (third column) basic pokemon.

.8833333333, .1166666667, 0
.7785310734, .2096045198, .0118644068
.6845704267, .2818819404, .0335476329
.6005003743, .3362802096, .0632194161
.5254378275, .3753127339, .0992494386
.4585639222, .4012434319, .1401926459
.3991204508, .4161042997, .1847752495
.3464064290, .4217121744, .2318813966
.2997747943, .4196847120, .2805404937
.2586292343, .4114556000, .3299151657
.2224211415, .3982890208, .3792898377
.1906466927, .3812933854, .4280599219
.1628440500, .3614343549, .4757215951
.1385906809, .3395471681, .5218621510
.1175007947, .3163482933, .5661509120
.09922289326, .2924464222, .6083306845
.08343743297, .2683528250, .6482097420
.06985459504, .2444910827, .6856543223
.05821216254, .2212062176, .7205816199
.04827350064, .1987732379, .7529532615
.03982563803, .1774051149, .7827692471
.03267744659, .1572602117, .8100623417

Some things to notice:

with only 8 basic pokemon (I've had some decks like this), you have a better than 1/3 chance of a mulligan, and less than 25% chance of having more than one basic pokemon at the start.
Even with 15 basic pokemon (I don't think I've played a deck with more than that many basics in some time) you have about one chance in 9 of a mulligan, and only a little better than a 50% chance of drawing more than one.
With 11 or fewer basics in the deck, you have a better chance of having only one basic pokemon in your draw than of having two or more.

Actually, the more I think about it, the more I think I should only list the probability of drawing zero pokemon as is -- what is relevant for the other two columns is their CONDITIONAL probability, given that you don't draw zero -- after all, if you draw zero, you shuffle and redraw. The real question ought to be -- how likely am I to start with at least one pokemon on the bench? But that's an easy fix to make, (I'd do it here, but this post is already too long).

[ July 20, 2002, 11:37 PM: Message edited by: Full_Heal_Energy ]

From: palo alto, california | Registered: Mar 2001 | IP: Logged

PsycoDad

Member # 40531

posted July 22, 2002 05:05 AM

I was taught by a "name" player how to shuffle. He used your method, Otaku! I have used it as well and it works. However, I now use the 6 pile blind shuffle and it also speads the cards out. In this method, you take your deck, put down 6 cards face down (seperated) on the table, and then take one card at a time from the top of the deck and stick it on top of one of the cards, making sure to put one card on each pile each round. You continue until the entire deck is in 6 piles. You can go in order or you can do the V pattern. I then offer the deck to my opponent and allow him to cut. If they shuffle, I cut when they are done. I hope this make sense.
[Eek!]

PsYcOdAd

[ July 22, 2002, 05:10 AM: Message edited by: PsycoDad ]

From: Shelby, Michigan | Registered: Aug 2001 | IP: Logged

Otaku

Member # 42359

posted July 22, 2002 01:56 PM

Crobat 1: Detasils please:) I am trying to figure out how you got that there would be a 20% of 2 being by each other. For some reason I can no longer seem to to proper probability anymore. I look at it like this: If I have a 60 card deck and with four Crobats and remove 1 of the Crobats, then stick it back in (all shuffling does is remove a variable amount of cards from the deck then stick them back in at random, so I feel it is safe to approach it from this angle. If I am wrong, tell me and explain, please [Smile]

) then there is a 3 in 59 (approximately 5.1%) chance of me sticking it by one of the other Crobats in the deck. 3 comes from the 3 Crobats left in the deck before I add the one I took out back, while 59 is of course the size of the deck without the Crobat in my hand. But I have not been having a good week mathematically speaking, so let me know if this is grossly inaccurate). [Wink]

FHE: Could you also explain the exact formula so I may use it in the future (I know I should know it...). Thanks.

Psycodad: As I mentioned earlier, I am now going to make the 10 piles and shuffle said piles back together without the pre-sorting. Thanks for sharing. [Smile]

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From: Iowa | Registered: Sep 2001 | IP: Logged

GreatFox

Member # 77642

posted July 22, 2002 04:42 PM

Well, the way I do it I see no problem with.

Way before a game, at home, I seperate the cards into piles of types, evos, energies and trainers, and kinda stack the deck. But before actually playing a game I might jave actually done about 25 or 50 quick blind rifle suffles to make sure that the deck is no longer stacked.

If you stack the deck before hand I don't really see a problem as long as you thourly rifle suffle the deck before a game. If you don't, then I would consider it cheating.

A good thing to do is on your way to a game, coninually rifle suffle your deck durring the trip to the game. You might end up suffling over 100 times before starting a game and thier should be no way that the deck is stacked afterwards.

But in no way would I suffle my deck in the way I described way above in this topic (I think its the third post) durring a game or even between games. Only at home would I do that.

--------------------
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From: Los Angeles, California | Registered: Apr 2002 | IP: Logged

Crobat1

Member # 85086

posted July 22, 2002 06:35 PM

Dear Otaku:

Will you be at Worlds? If so, it is much easier to explain these kinds of problems interactively.

I will try--briefly.

First of all, if you had removed a Crobat and then placed it back it, it would have a greater chance than you realize of touching one of the other 3 Crobats. Assuming that the 3 Crobats were not touching, not one space away from each other, and not on the top and bottom, there would be 60 possible places for the new Crobat and 6 spots that would touch one of the others--that is, a 10% chance. The 3 contingencies which I mentioned must be taken into account. More importantly, there is a significant chance that the original 3 Crobats are already touching.

Here is how I approach the problem.

Think of the 60 cards as existing in a circle (that is, temporarily eliminate the possibility of a top and a bottom to the deck). Now number the Crobats, 1 through 4. Any pair of Crobats have a 2 in 59 chance of touching each other. There are 6 pair of Crobats (1-2, 1-3, 1-4, 2-3, 2-4, 3-4). Thus I come up with the approximation that 20% of the time, 2 of them will be touching. 12 in 59 would be a better starting number, but one has to reduce this number by 1/60 in order to take into account that the two "touching" Crobats may be on the top and bottom of the deck. This brings us to 12 in 59 multiplied by 59 in 60, which brings us to 20%.

More precisely, although it is true that in 100 randomly assorted decks, there will be an average of 20 pair of Crobats touching each other, one has to take into account that 2 of these touching pairs may occur in the same deck, which is why I said approximately 20%. My offhand guess is that it is closer to 19%. (Please note that 3 Crobats next to each other counts as 2 pair of touching Crobats in this logic and all four next to each other counts as 3 touching pairs.)

There may be a simpler way of expalining this--but I don't know it.

Hope to see ya at Worlds.

--------------------
Bilbo Baggins: "Every worm has its weakness."

Crobat2:
Winner, Syracuse SBZ, Feb, 2003
22nd Place, 10 and Under, World's, August, 2002
Winner, Friday's Open Modified Event, Origins, July, 2002
Winner, 10 and Under Gym Challenge, Milford, May, 2002

Crobat1:
Winner, Albany SBZ, May, 2003
9th Place: Professor's Tournament at World's, August 2002
Top 4, Theme Deck Tournament at World's, August, 2002
Top 8, Unlimited Side Event at World's (Undefeated in Main Draw), August, 2002

From: Binghamton, NY | Registered: May 2002 | IP: Logged

Full_Heal_Energy

Member # 10225

posted July 22, 2002 09:47 PM

I'll write a longer post after I put the kids to bed, but right now I'll say:

the odds two (or more) crobats are next to each other if you have four in your 60 card deck is:

1 - (57*56*55*54)/(60*59*58*57)
= .1899473992
Crobat1's approximation is good, and his estimate of the error is VERY good.

I might also write this as
1 - binomial(57,4)/binomial(60,4)
(where binomial(n,k) means n!/( k! * (n-k)!) and
"!" stands for "factorial")

if you have n of a certain type of card (like basic pokemon) in your 60 card deck and you
draw 7 cards, the odds of getting exactly k of
that type among the 7 is:

binomial(n,k)*binomial(60-n,7-k) / binomial(60,7)

honest! After I put the kids to bed, I'll try to write up some explanations for these formulas.

[Incidentally, even though I know all this math, you'll probably beat me if we play at Worlds...]

[ July 22, 2002, 10:02 PM: Message edited by: Full_Heal_Energy ]

From: palo alto, california | Registered: Mar 2001 | IP: Logged

Full_Heal_Energy

Member # 10225

posted July 22, 2002 11:06 PM

OK, here are two ways to think about the four crobat problem.

This first one may be the most straightforward way to think about it -- though it is not the most elegant:

you have 56 other cards -- say they're all grass energy -- and four crobats. [I know, I know, you need zubats, and golbats or breeders too, but bear with me ...]

start with all 56 grass energy and one crobat, all shuffled nicely. There are 57 cards in the deck -- which means there are 58 places we could randomly add the next card (either at the top of the deck, or after any one of the 57 cards, including the last card) Of those 58 locations, TWO of them are next to the one crobat already in the deck (either just above the crobat or just below it)

So if you randomly add one more crobat, you have 2/58 chance of putting it next to the existing crobat and 56/58 chance of NOT doing so.

Now, when it winds up next to the first crobat, game over... but the other 56/58 of the time, you now have two crobats in the deck, NOT next to each other. So there are 58 cards in the deck, 59 places we could randomly add the next card and FOUR of these places are next to crobats. [note since the crobats already in the deck aren't adjacent, we aren't double counting any of the places] So you have 4/59 chance of putting a randomly inserted card next to a crobat and 55/59 chance of NOT doing so.

So, the probability that you could put up to 3 crobats randomly in a 59 card deck without getting any two adjacent is:
56/58 * 55/59 .

And, if you are lucky enough to still have no crobats next to each other -- continue: There are now 59 cards in the deck, 60 locations for the next crobat, and SIX of those are adjacent to one of the other crobats... so you have a 54/60 chance that you will NOT place it next to a crobat. This means that the chance of having four crobats in a 60 card deck with no two adjacent is
56/58 * 55/59 * 54/60

and so the chance that, in adding those cards, that you DID get two [or more] adjacent at one of those steps along the way, is 1 - (56*55*54)/(58*59*60) = .1899473992

This is equivalent to the formula I gave before, but in a slightly different format.

Here's another way to look at it -- but it's a little more abstract, and needs the fact:

quote:

The number of different subsets of size k taken from a collection of n objects is binomial(n,k)

(proof on request, but this may already be too long and abstract.)

Then:
there are binomial(60,4) ways to pick four locations for the crobats in the deck. All of them are equally likely.

However, how many of them do not have two or more crobats next to each other? It turns out to be binomial(57,4) -- why?

Well, there's a one-to-one correspondence between the possible locations for four crobats into a deck with a total of 57 cards (4 crobats and 53 grass energys) -- with, however, the crobats possibly adjacent -- and the number of possible locations for four crobats in a 60 card deck with no two adjacent.

[You can see the correspondence this way: if you start with the 4 from 57, just thrust an extra grass energy into each of the three gaps between crobats. If you start with the 4 from 60 with no two crobats adjacent, just take one of the grass energys FROM each of the gaps (they all must have at least one). This really is one-to-one. Every possible way to have four crobats and 53 grass energy is paired with exactly one way to have 4 crobats with no two adjacent and 56 grass energy.]

So the number of possible locations for four crobats in a 60 card deck, with no two adjacent is binomial(57,4) . This means the chances of getting no two together in a random 60 card deck is binomial(57,4)/binomial(60,4). And thus, the chances of getting two together is:
1 - binomial(57,4)/binomial(60,4) which again is .1899473992

[ July 22, 2002, 11:09 PM: Message edited by: Full_Heal_Energy ]

From: palo alto, california | Registered: Mar 2001 | IP: Logged

Pokidad

Member # 135

posted July 23, 2002 06:43 AM

Agggghhhh. You guys are going to make me take out my Business Statistics Books and look up the formulas of determining events and their probability when something to the "n" degree. I love math and shapes of curves. Ooops, I hope "Da Ladies Man (aka Sensei) didn't here that. [ROFL]

Actually, this is a great discussion and excellent statistics problem for determining random events and probability that the next card or group of cards will contain a particular card. For those who love to learn a little more about statistics, please look up FHE and Crobat1, I know I will.

Guys, please have one of the MTs point me out to you at Worlds. I have the 8:30 - 12:30 deck check shift on Saturday and would love to meet you guys. I'll try to brush up on my statistics for your class.

Now, this would be a fun course to try to teach to the kids, in their language. [Big Grin]

--------------------
What happen to my Sneasel? See what happens when you let him out of the box.

From: Falls Church, VA USA | Registered: Feb 2001 | IP: Logged

Crobat1

Member # 85086

posted July 23, 2002 12:42 PM

Dear Pokidad:

I look forward to meeting you. Thank you for responding to my son's question about how language is dealt with at the Worlds.

As for statistics, remember, the only real important statistic is that Crobat confuses opponents 11 out of 16 times. My numbers are probably higher. On a good day, I can confuse everyone. The fourth row of Pascal's triangle is 1-4-6-4-1, which tells you the probabilities attached to randomly flipping 4 coins.

I personally find that Pascal's triangle is enormously helpful in determining the record that will be needed to make it into the playoffs. Kids love it, my 10 year old included.

See ya at Worlds.

--------------------
Bilbo Baggins: "Every worm has its weakness."

Crobat2:
Winner, Syracuse SBZ, Feb, 2003
22nd Place, 10 and Under, World's, August, 2002
Winner, Friday's Open Modified Event, Origins, July, 2002
Winner, 10 and Under Gym Challenge, Milford, May, 2002

Crobat1:
Winner, Albany SBZ, May, 2003
9th Place: Professor's Tournament at World's, August 2002
Top 4, Theme Deck Tournament at World's, August, 2002
Top 8, Unlimited Side Event at World's (Undefeated in Main Draw), August, 2002

From: Binghamton, NY | Registered: May 2002 | IP: Logged

Crobat1

Member # 85086

posted July 23, 2002 01:38 PM

First: everyone except for Full Heal Energy should not read this--or suffer the consequence--mathematicitus.

Full Heal Energy: I love probability problems, but tend to avoid formulas for these kinds of problems--because they can easily go astray. I have generally tried to wing it--at my own peril.

Your first more practical (less formulaic) solution is more readable and more compelling--but it is slightly off.

I agree with you that inserting a second Crobat into a deck with 57 cards (including 1 Crobat) has a 2 out of 58 chance of leading to 2 Crobats touching.

Once, however, there are already 2 Crobats in the deck, the probability of inserting a 3rd Crobat which touches one of the 1st two Crobats is actually slightly less than 4 out of 59. Why? There is a possibility that the first 2 Crobats are in the deck with only one card between them, in which case, the probability of placing a 3rd Crobat into the deck and touching one of the first 2 is actually 3 out of 59. Similarly, the probability of inserting a 4th Crobat into a deck and touching one of the 1st 3 Crobats is slightly less than 6 out of 60. What I cannot figure out is how your second solution provided the same numerical answer as your first method.

What we agree about is that shuffling decks does not lead to decks that have no similar cards touching, which is why so many people are tempted to want to find a way of more reliably insuring that their decks are more consistently diversified.

I look forward to meeting you at Worlds.

Crobat1

--------------------
Bilbo Baggins: "Every worm has its weakness."

Crobat2:
Winner, Syracuse SBZ, Feb, 2003
22nd Place, 10 and Under, World's, August, 2002
Winner, Friday's Open Modified Event, Origins, July, 2002
Winner, 10 and Under Gym Challenge, Milford, May, 2002

Crobat1:
Winner, Albany SBZ, May, 2003
9th Place: Professor's Tournament at World's, August 2002
Top 4, Theme Deck Tournament at World's, August, 2002
Top 8, Unlimited Side Event at World's (Undefeated in Main Draw), August, 2002

From: Binghamton, NY | Registered: May 2002 | IP: Logged

Full_Heal_Energy

Member # 10225

posted July 23, 2002 03:04 PM

Crobat1 said:
>Your first more practical (less formulaic)
>solution is more readable and more compelling--
>but it is slightly off.

No, it isn't -- let me explain

>I agree with you that inserting a second Crobat
>into a deck with 57 cards (including 1 Crobat)
>has a 2 out of 58 chance of leading to 2 Crobats
>touching.

yes.
>Once, however, there are already 2 Crobats in
>the deck, the probability of inserting a 3rd
>Crobat which touches one of the 1st two Crobats
>is actually slightly less than 4 out of 59.
No.

>Why? There is a possibility that the first 2
>Crobats are in the deck with only one card
>between them, in which case, the probability of
>placing a 3rd Crobat into the deck and touching
>one of the first 2 is actually 3 out of 59.

No, this is wrong -- there are two spaces between
the two crobats -- one below the top crobat
and above the non-crobat and one below the non-crobat and above the bottom crobat. Added to the
space above the top crobat and the one below the bottom crobat makes four.

>Similarly, the probability of inserting a 4th >Crobat into a deck and touching one of the 1st 3
>Crobats is slightly less than 6 out of 60.

Nope -- when the crobats are NOT already adjacent,
they don't share any of the spaces between cards.

>What I cannot figure out is how your second
>solution provided the same numerical answer as
>your first method.

They are equivalent.

But we can talk about all this at Worlds -- I'm looking forward to meeting everyone. But I have to warn Pokidad that while you might be able to use Pokemon to introduce kids to math, you have to worry about using math to scare kids away from Pokemon....

From: palo alto, california | Registered: Mar 2001 | IP: Logged

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