Unlimited Base Set Booster Box

[bailster]

@squirtle1000

@bailster ,

In this example there’s only one box being opened which we assume has 12 holo packs. 61.6% is the probability of getting at least 1 Charizard in the box, based on the above assumption. You could get 1, 2, or 3 Charizards. The total probability of getting either 1, 2, or 3 Charizards is 61.6%.

Yes but is the assumption statistically relevant? A assumption that happens 1 in 100 boxes isnt relevant.

[squirtle1000]

@bailster

@squirtle1000

@bailster ,

In this example there’s only one box being opened which we assume has 12 holo packs. 61.6% is the probability of getting at least 1 Charizard in the box, based on the above assumption. You could get 1, 2, or 3 Charizards. The total probability of getting either 1, 2, or 3 Charizards is 61.6%.

Yes but is the assumption statistically relevant? A assumption that happens 1 in 100 boxes isnt relevant.

There’s two possiblities I’ve considered:

1. Wizards/TPC specifically made it so that you can only get at most 3 of a particular holo in a box. I don’t know if they intentionally did this, or if it is even true. For the sake of doing the calculation, I made the assumption. Gary or others could chime in and say whether they ever received 4 copies of the same holo in a box. If a limit of 3 is the reality then the odds of 1 or more Charizards in a box is 61.6%.

2. The other possibility I considered is there is no restriction, and Wizards/TPC made it so you could theoretically get 12 Nidokings in a box, if you were lucky (or unlucky, depending on your perspective). If this is theoretically possible, then the odds of 1 or more Charizards in a box is 56.3%.

[zorloth]

@squirtle1000

@bailster

Yes but is the assumption statistically relevant? A assumption that happens 1 in 100 boxes isnt relevant.

There’s two possiblities I’ve considered:

1. Wizards/TPC specifically made it so that you can only get at most 3 of a particular holo in a box. I don’t know if they intentionally did this, or if it is even true. For the sake of doing the calculation, I made the assumption. Gary or others could chime in and say whether they ever received 4 copies of the same holo in a box. If a limit of 3 is the reality then the odds of 1 or more Charizards in a box is 61.6%.

2. The other possibility I considered is there is no restriction, and Wizards/TPC made it so you could theoretically get 12 Nidokings in a box, if you were lucky (or unlucky, depending on your perspective). If this is theoretically possible, then the odds of 1 or more Charizards in a box is 56.3%.

Another possibility (the most likely one) is that the distribution of holos within the 12 holo packs of a given Base set box isn’t binomial. Holos are cut from a sheet and each sheet contains an equal quantity of each holo. These holos are then machine-inserted along with the other cards into packs, and the packs are then machine-collated into booster boxes. Each pack in a booster box doesn’t give an equal chance at pulling a specified holo. Which holo is pulled from one pack influences the likelihood of pulling a specified holo from another pack in that box – these aren’t independent trials.

If you’re still not convinced, take a look at box mapping in MTG. There are predictable patterns of distribution of cards of a specific rarity within booster boxes. The presence or absence of specific rares in specific locations makes near certain the presence or absence of other rares in other specified locations, and so on with each rarity level.

[squirtle1000]

@zorloth

@squirtle1000

There’s two possiblities I’ve considered:

1. Wizards/TPC specifically made it so that you can only get at most 3 of a particular holo in a box. I don’t know if they intentionally did this, or if it is even true. For the sake of doing the calculation, I made the assumption. Gary or others could chime in and say whether they ever received 4 copies of the same holo in a box. If a limit of 3 is the reality then the odds of 1 or more Charizards in a box is 61.6%.

2. The other possibility I considered is there is no restriction, and Wizards/TPC made it so you could theoretically get 12 Nidokings in a box, if you were lucky (or unlucky, depending on your perspective). If this is theoretically possible, then the odds of 1 or more Charizards in a box is 56.3%.

Another possibility (the most likely one) is that the distribution of holos within the 12 holo packs of a given Base set box isn’t binomial because the trials aren’t independent. Holos are cut from a sheet and each sheet contains an equal quantity of each holo. These holos are then machine-inserted along with the other cards into packs, and the packs are then collated into booster boxes. It’s exceptionally unlikely that there’s a binomial distribution of holos within each booster box.

If you’re still not convinced, take a look at box mapping in MTG. There are predictable patterns of distribution of cards of a specific rarity within booster boxes. The presence or absence of specific rares in specific locations makes near certain the presence or absence of other rares in specific locations, and so on with each rarity level.

Yes, I think you are probably correct. However, I’d still argue that unless Wizards specifically made certain base holos rarer than others, then otherwise in an ensemble of 1000 booster boxes manufactured at arbitrary times over the course of the print run, if one were to open all of them one would probably find that 1 or more Charizards showed up in somewhere around 6 out of 10 boxes, as opposed to 8 out of 10.

[zorloth]

@squirtle1000

@zorloth

Another possibility (the most likely one) is that the distribution of holos within the 12 holo packs of a given Base set box isn’t binomial because the trials aren’t independent. Holos are cut from a sheet and each sheet contains an equal quantity of each holo. These holos are then machine-inserted along with the other cards into packs, and the packs are then collated into booster boxes. It’s exceptionally unlikely that there’s a binomial distribution of holos within each booster box.

If you’re still not convinced, take a look at box mapping in MTG. There are predictable patterns of distribution of cards of a specific rarity within booster boxes. The presence or absence of specific rares in specific locations makes near certain the presence or absence of other rares in specific locations, and so on with each rarity level.

Yes, I think you are probably correct. However, I’d still argue that unless Wizards specifically made certain base holos rarer than others, then otherwise in an ensemble of 1000 booster boxes manufactured at arbitrary times over the course of the print run, if one were to open all of them one would probably find that 1 or more Charizards showed up in somewhere around 6 out of 10 boxes, as opposed to 8 out of 10.

~6/10 would be the probability if each holo pack were an independent trial with a success rate of 1/15. But it seems likely that WotC ensured a more equitable distribution of Charizards (and each holo) across boxes. So I wouldn’t be surprised if the actual probability of opening one or more Charizards out of a Base Set box was closer to 80% than 60%.

[squirtle1000]

@zorloth

@squirtle1000

Yes, I think you are probably correct. However, I’d still argue that unless Wizards specifically made certain base holos rarer than others, then otherwise in an ensemble of 1000 booster boxes manufactured at arbitrary times over the course of the print run, if one were to open all of them one would probably find that 1 or more Charizards showed up in somewhere around 6 out of 10 boxes, as opposed to 8 out of 10.

~6/10 would be the probability if each holo pack were an independent trial with a success rate of 1/15. But it seems likely that WotC ensured a more equitable distribution of Charizards (and each holo) across boxes. So I wouldn’t be surprised if the actual probability of opening one or more Charizards out of a Base Set box was closer to 80% than 60%.

Hmm. I agree that they would have ensured an equitable distribution across boxes. However, I still think only ~6/10 boxes would contain one or more Charizards. Emphasis on that phrase. The expectation value would be 8 Charizards in 10 boxes. That doesn’t translate to an 80% probability of getting a Charizard in a particular box, since some boxes would have 2 or 3, which help offset for the other boxes which had 0.

So, I still think my argument holds for an ensemble of tons of boxes, if one is simply asking “Hey, about what percent of boxes did you get at least one Charizard hit?”. If, however, you ask how many Charizards a person got out of 1000 boxes, the answer is quite likely near 800.

Put another way:
“Perhaps 6 in 10 boxes have at least one Charizard, but you’ll probably get a total of 8 Charizards in 10 boxes, because in 1 or 2 of those boxes you got lucky and got more than 1 Charizard.”

[KingPokemon]

@squirtle1000

@KingPokemon

If there’s 12 holos in a box, and 15 unique ones, wouldn’t that make it an 80% chance to pull any specific one?

Intuitively it seems like it, but it doesn’t quite work that way.

What you are saying would be true IF each box had a guarantee that you wouldn’t get duplicate holos. If there were no duplicates then in that case you’d get 12 unique holo Pokemon in the 12 holo packs. Since there are only 15 Pokemon to choose from, there’d be an 80% chance that Charizard would be one of your 12 unique pulls. That’s probably why it’s easy to assume there’s an 80% chance of getting Charizard.

Another way to see why this isn’t true is imagine that Base Set was smaller and the packs had only 10 different holo Pokemon to choose from, instead of 15. Then you might be tempted to say there’s 10 unique holos, 12 packs, so 120% chance you’d get a specific one. So guaranteed. Of course that can’t be true, because you might get a ton of duplicates and still not get a specific one you are after.

So yeah, the calculations I showed above give the answer according to probability theory.

P = Probability of at least 1 copy of a specific holo in k packs, if there are N unique holos in the set:

P = 1 - ( (N-1) / (N) )^k

k = 12 for every booster box
N = 15 for base set

But you also might get multiple Charizards?

[squirtle1000]

@KingPokemon

@squirtle1000

Intuitively it seems like it, but it doesn’t quite work that way.

What you are saying would be true IF each box had a guarantee that you wouldn’t get duplicate holos. If there were no duplicates then in that case you’d get 12 unique holo Pokemon in the 12 holo packs. Since there are only 15 Pokemon to choose from, there’d be an 80% chance that Charizard would be one of your 12 unique pulls. That’s probably why it’s easy to assume there’s an 80% chance of getting Charizard.

Another way to see why this isn’t true is imagine that Base Set was smaller and the packs had only 10 different holo Pokemon to choose from, instead of 15. Then you might be tempted to say there’s 10 unique holos, 12 packs, so 120% chance you’d get a specific one. So guaranteed. Of course that can’t be true, because you might get a ton of duplicates and still not get a specific one you are after.

So yeah, the calculations I showed above give the answer according to probability theory.

P = Probability of at least 1 copy of a specific holo in k packs, if there are N unique holos in the set:

P = 1 - ( (N-1) / (N) )^k

k = 12 for every booster box
N = 15 for base set

It you also might get multiple Charizards?

@KingPokemon,

Yeah, getting multiple is factored into that. I was specifically saying what the odds might be for getting a box that has at least 1 Charizard. That includes the cases where you’d get more than 1.

I didn’t specifically calculate the odds of getting exactly 2, or exactly 3, since I was just interested in figuring out the odds of any vs. none.

This has spread across a lot of posts, so my summary of everything is:

Statistically, 6 out of 10 boxes will give you at least 1 Charizard hit. 4 out of 10 boxes won’t give you any. You’ll probably get 8 Charizards in those 10 boxes, which is likely what most people would have guessed before all this, anyway. So the only possibly new information here is the idea that maybe 40% of boxes are expected to have none. Of course that’s balanced out by the fact that some boxes have 2 or 3, hence the 8 Charizards expected in 10 boxes.

[zorloth]

@squirtle1000

@zorloth

~6/10 would be the probability if each holo pack were an independent trial with a success rate of 1/15. But it seems likely that WotC ensured a more equitable distribution of Charizards (and each holo) across boxes. So I wouldn’t be surprised if the actual probability of opening one or more Charizards out of a Base Set box was closer to 80% than 60%.

Hmm. I agree that they would have ensured an equitable distribution across boxes. However, I still think only ~6/10 boxes would contain one or more Charizards. Emphasis on that phrase. The expectation value would be 8 Charizards in 10 boxes. That doesn’t translate to an 80% probability of getting a Charizard in a particular box, since some boxes would have 2 or 3, which help offset for the other boxes which had 0.

So, I still think my argument holds for an ensemble of tons of boxes, if one is simply asking “Hey, about what percent of boxes did you get at least one Charizard hit?”. If, however, you ask how many Charizards a person got out of 1000 boxes, the answer is quite likely near 800.

Put another way:
“Perhaps 6 in 10 boxes have at least one Charizard, but you’ll probably get a total of 8 Charizards in 10 boxes, because in 1 or 2 of those boxes you got lucky and got more than 1 Charizard.”

I understand that; I’m also talking about boxes that contain one or more. My point is that the more equitable distribution could have easily resulted in a closer to 80% than 60% probability that at least one Charizard is pulled out of a given Base Set booster box. After all, an entirely random distribution already results in a 56% chance of pulling at least one Charizard in a given box. So I don’t think it’s farfetched that a distribution more equitable than a random distribution could result in in a close-to-80% chance (which would mean, of course, that few boxes contain more than one Charizard).

[squirtle1000]

@zorloth,

Ah, right. I see where you are coming from. Yeah I guess it all depends on how equitable they made it. Like you said, the more equitable it is, the less likely it is to have boxes with 2 or 3 Charizards, so it does go both ways to flatten the distribution.

The data probably exists on YouTube to answer this rather definitively. There must be well over 100 Base Unlimited box breaks uploaded. It would just be a matter of watching them all and tallying the results, which I definitely don’t feel like doing!

[KingPokemon]

@squirtle1000

@zorloth,

Ah, right. I see where you are coming from. Yeah I guess it all depends on how equitable they made it. Like you said, the more equitable it is, the less likely it is to have boxes with 2 or 3 Charizards, so it does go both ways to flatten the distribution.

The data probably exists on YouTube to answer this rather definitively. There must be well over 100 Base Unlimited box breaks uploaded. It would just be a matter of watching them all and tallying the results, which I definitely don’t feel like doing!

So it’s decided your 56% Charizard is no longer true?

[squirtle1000]

@KingPokemon

@squirtle1000

@zorloth ,

Ah, right. I see where you are coming from. Yeah I guess it all depends on how equitable they made it. Like you said, the more equitable it is, the less likely it is to have boxes with 2 or 3 Charizards, so it does go both ways to flatten the distribution.

The data probably exists on YouTube to answer this rather definitively. There must be well over 100 Base Unlimited box breaks uploaded. It would just be a matter of watching them all and tallying the results, which I definitely don’t feel like doing!

So it’s decided your 56% Charizard is no longer true?

I don’t think there’s enough data to say definitively. There’s two extremes, 56% and 80%. We know the correct value is somewhere in that range.

If holos are tossed in packs completely randomly with no attempt to suppress duplicates, the answer could be as low as 56%.
If duplicate holos were never pulled from packs, the answer would certainly be 80%. This we know isn’t true, there are duplicate pulls.

So the real answer is somewhere between 56-80%, depending on how balanced Wizards tried to be with their boxes. That is, how far they departed from pure randomness. I’m inclined to say it’s still under 70%, based on how often we see duplicates in box pulls. Zorloth would probably say it’s over 70%.
We won’t know the answer with any confidence until someone watches every box break and shares the stats!

[KingPokemon]

I’m gonna throw another wrench into the works.
There were fewer of some cards produced than others. For example, all English base set holo sheets are identical containing 111 cards. Some had 7 on the sheet, some had 8. Charizards had 7. Chansey had 8. And so on.
Wanna try your calculations again;)

[seafoamarticuno]

@KingPokemon

I’m gonna throw another wrench into the works.
There were fewer of some cards produced than others. For example, all English base set holo sheets are identical containing 111 cards. Some had 7 on the sheet, some had 8. Charizards had 7. Chansey had 8. And so on.
Wanna try your calculations again;)

One thing I’ve always wondered is if Theme Deck holos come from the normal Holo sheets or from special theme deck only sheets… This could also change the odds. Anyone know one way or the other?

[squirtle1000]

@KingPokemon

I’m gonna throw another wrench into the works.
There were fewer of some cards produced than others. For example, all English base set holo sheets are identical containing 111 cards. Some had 7 on the sheet, some had 8. Charizards had 7. Chansey had 8. And so on.
Wanna try your calculations again;)

Very interesting!

Assuming theme deck holos were cut from their own sheets, and a sheet of 111 went purely into booster packs, then that means Charizard is slightly rarer. We would no longer be able to say 1/15 = 0.067% of booster pack holos ever produced is Charizard. Instead, it would be 7/111 = 0.063%.

I did some quick calculations, and I think the new range would be 54.2% - 75.7%, instead of 56.3% - 80%.

Anyway, yeah this is going to entirely depend on how theme deck holos were obtained.

[KingPokemon]

Theme decks had their own sheets.

[squirtle1000]

@garyis2000,

Do you know if Venusaur and Blastoise also had only 7 copies per sheet?

[KingPokemon]

[squirtle1000]

I was bored so I added them up:

Venusaur: 7
Ninetales: 7
Gyarados: 7
Mewtwo: 7
Hitmonchan: 7
Raichu: 7
Charizard: 7
Blastoise: 7
Clefairy: 7
Alakazam: 7
Zapdos: 8
Chansey: 8
Poliwrath: 8
Magneton: 8
Nidoking: 8

This is almost exactly what I would have expected. I always felt like Magneton, Poliwrath, Chansey, and Nidoking were a little more common than the others. The only one I wouldn’t have guessed was Zapdos. Maybe I’m biased - Zapdos was my first ever holo pull from a booster pack.

Notice that they chose to keep all 4 theme deck holos a little more rare in packs. That makes a lot of sense, since they knew there would be a lot of those theme deck holos in circulation. Plus they kept the starters at that higher rarity too, which also makes sense.

[KingPokemon]

@squirtle1000,
Something you might find interesting, I pulled 3 1st base charizards once (with two witnesses who are still alive;) That was waaaay harder than the 3 Chancey’s Logan pulled out of his box recently.