So I opened 2 packs... (Maths Question)

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Hey all

Last night I opened two RTR packs, and got:
1 x Foil Firemind's Foresight and 1 x Firemind's Foresight in one pack
and:
1 x Firemind's Foresight in the other pack!

The more I started thinking about it, the more complicated it seemed to be to work out the odds of that happening, so I thought I would pose the question here:

Q: What are the odds of getting 3 matching rares (inc. mythic rares) in two packs?
(not necessarily a particular rare, just rares that match)

~ Tim
I am Blue/White Reached DCI Rating 1800 on 28/10/11. :D
Sig
56287226 wrote:
190106923 wrote:
Not bad. But what happens flavor wise when one kamahl kills the other one?
Zis iz a sign uf deep psychological troma, buried in zer subconscious mind. By keelink himzelf, Kamahl iz physically expressink hiz feelinks uf self-disgust ova hiz desire for hiz muzzer. [/GermanPsychologistVoice]
56957928 wrote:
57799958 wrote:
That makes no sense to me. If they spelled the ability out on the card in full then it would not be allowed in a mono-black Commander deck, but because they used a keyword to save space it is allowed? ~ Tim
Yup, just like you can have Birds of paradise in a mono green deck but not Noble Hierarch. YAY COLOR IDENTITY
56287226 wrote:
56888618 wrote:
Is algebra really that difficult?
Survey says yes.
56883218 wrote:
57799958 wrote:
You want to make a milky drink. You squeeze a cow.
I love this description. Like the cows are sponges filled with milk. I can see it all Nick Parks claymation-style with the cow's eyes bugging out momentarily as a giant farmer squeezes it like a squeaky dog toy, and milk shoots out of it.
56287226 wrote:
56735468 wrote:
And no judge will ever give you a game loss for playing snow covered lands.
I now have a new goal in life. ;)
Low. I'm dreadfully sorry, btw. We at least the uncommons good? It's a shame you weren't drafting with those.
76783093 wrote:
Luckily, we have stop-having-fun guys to remind us that having anything more than 60 cards in your deck is tantamount to being a rapist and anyone considering it should be strung up by their ****.
Ok, the odds of getting 1 regular  Firemind's Foresight will be the probability of getting a rare multiplied by the probability of getting the specific rare, Firemind's Foresight (or any other specific rare for that matter).

If we assume that every 1 in 10 packs contains a Mythic, then every 9 in 10 packs contains a Rare.  Assuming all rares are equally likely, the probability of getting Firemind's Foresight is 1 in 53.

Therefore, the probability of getting a regular Firemind's Foresight is

0.9 x 1/53 = 0.017 or 1.7% chance.

The probability of getting a foil Firemind's Foresight will equal the probability of getting a foil multiplied by the probability that the foil is the specific card, Firemind's Foresight.  I'll make the assumption that every 1 out of 10 packs contains a foil (I have no idea what the actual distribution of foils is).  Also, I'll assume that all cards are equally likely (again, when it comes to the foils, I have no idea how rareity comes into play whith how foils are distributed).  Then, the odds of getting the specific foil, Firemind's Foresight, is

0.1 x 1/254 = 0.0003937 or  about a 0.04% chance.

Ok, this is where it gets a little complicated.  The event we're looking for is the probability that the first pack contains a regular and foil Firemind's Foresight and the second pack contains only a regular Firemind's Foresight -- OR -- (to answer the question in bold) the first pack contains only a regular Firemind's Foresight and the second pack contains both a regular and foil Firemind's Foresight.  Since what happens in pack 1 has no influence on what happens in pack 2, we can simply multiply the individual probabilities.  Breaking that down...

Odds = Prob(pack 1 = regular FF and foil FF)*Prob(pack 2 = regular FF and not foil FF) +...
      Prob(pack 1 = regular FF and not foil FF)*Prob(pack 2 = regular FF and foil FF)

The probability of gettng both a foil and regular FF = the two probabilities multiplied together.

0.017 x 0.0003937 = 0.0000066929

The probability of getting a regular FF and not a foil FF, similarly is

0.017 x (1 - 0.0003937) = 0.0170.  (The second term is so close to 1 that the probability is practically the same as getting a regular FF and not caring whether or not you get a foil one.)

Plug these in the equation for Odds...

Odds = (0.0000066929)*(0.0170) + (0.0170)*(0.0000066929) = 0.00000022756.

This is about a 1 in 4,394,500 chance that you would get three of the same rare in two packs.  Again, I don't know what the actual distribution of foils is, and I suspect it's probably a little higher than 1 in 10.  But, you get the idea.
Mythics are one in eight packs, not ten.

The frequency of foils has never been published to my knowledge and may vary between sets, but MTGS Wiki claims it's around one in seventy packs.
blah blah metal lyrics
Mythics are one in eight packs, not ten.

The frequency of foils has never been published to my knowledge and may vary between sets, but MTGS Wiki claims it's around one in seventy packs.

It is published in the booster wrap itself; about 1:67 cards in most sets are premium/foil.

[<o>]
I think the odds are very high.  That, search the city, guild fued, death's presence, and jarard's orders are the only rares in this set.  At least from opening hunderds of packs in draft.  Other people around me keep smiling and saying something about some dual land or other things they open but I'm pretty sure they are liars or just need glasses.
I opened a Nicol Bolas, Planeswalker in a pack, and then opened a second one in a second pack. If your packs came from loose packs the odds are higher from the little I know than if you pulled them from two adjacent packs out of a sealed booster box.

Please check out my Blog:

Magic the Gathering Adventures Blog

http://mtgadventures.blogspot.com/

Please check out my YouTube channel:

http://www.youtube.com/user/rubiera22/featured

 

Mythics are one in eight packs, not ten.


Prob(rare in pack) = 7/8
It is published in the booster wrap itself; about 1:67 cards in most sets are premium/foil.


Given Prob(card is foil) = 1 foil/67 cards, use a little unit conversion...
(1 foil/67 cards)*(15 cards/pack) yields Prob(foil in pack) = 0.2239 or somewhere between every 4th or 5th pack.  From experience, that seems about right.

Adjust the math...
Prob(regular FF and foil FF in pack) = 0.000014552
Prob(regular FF and not foil FF in pack) = 0.0165

Odds = (0.000014552)(0.0165) + (0.0165)(0.000014552) = 0.00000048006

Which is about a 1 in 2,083,100 chance you would get three of the same rare in two packs.  Again, this assumes all cards equally likely in foil (because I couldn't find any info on how rareitites are distributed among foils).

I'll give it a bash, although I haven't used probability seriously for around ten or more years....

Probability of getting a rare in the first pack = 7/8
Probability of a matching rare in pack two = 7/8 x 1/53 = 7/424
Probability of a matching foil in either pack = 1/67*30 x 0.875/15 x1/53 = 26.25/53265
Probability of all together = 7/8 x 7/424 x 26.25/53265 =  1286.25/180674880 = Appx 1/140466

I assumed that the distribution of rarity through foils is the same as regular cards.

Cheers

EDIT: This doesn't exlude the possibility of a second matching foil rare.
Probability of getting a rare in the first pack = 7/8


I assumed a specific rare, but I think your right here, we don't necessarily care which rare in the first pack.
Probability of getting a rare in the first pack = 7/8


I assumed a specific rare, but I think your right here, we don't necessarily care which rare in the first pack.

Yea, it definitely depends on whether your answering the odds of three Firemind's Foresight or the odds of three of any rare.  I'm not sure if my foil workings are correct though.

Cheers
Low. I'm dreadfully sorry, btw. We at least the uncommons good? It's a shame you weren't drafting with those.


I was *supposed* to be keeping them for Packwars, but I had a rough day and thought I would take a peak "to cheer me up". Suffice to say it didnt work (it did freak me out a bit though...).

The commons and uncommons were junk too.

Best thing about the packs was the 8/8 Vigilance Token which I had struggled to get since the set came out... until last week, when I traded for one off a friend. :/

Freaky packs.

Lesson learnt.

~ Tim 

I am Blue/White Reached DCI Rating 1800 on 28/10/11. :D
Sig
56287226 wrote:
190106923 wrote:
Not bad. But what happens flavor wise when one kamahl kills the other one?
Zis iz a sign uf deep psychological troma, buried in zer subconscious mind. By keelink himzelf, Kamahl iz physically expressink hiz feelinks uf self-disgust ova hiz desire for hiz muzzer. [/GermanPsychologistVoice]
56957928 wrote:
57799958 wrote:
That makes no sense to me. If they spelled the ability out on the card in full then it would not be allowed in a mono-black Commander deck, but because they used a keyword to save space it is allowed? ~ Tim
Yup, just like you can have Birds of paradise in a mono green deck but not Noble Hierarch. YAY COLOR IDENTITY
56287226 wrote:
56888618 wrote:
Is algebra really that difficult?
Survey says yes.
56883218 wrote:
57799958 wrote:
You want to make a milky drink. You squeeze a cow.
I love this description. Like the cows are sponges filled with milk. I can see it all Nick Parks claymation-style with the cow's eyes bugging out momentarily as a giant farmer squeezes it like a squeaky dog toy, and milk shoots out of it.
56287226 wrote:
56735468 wrote:
And no judge will ever give you a game loss for playing snow covered lands.
I now have a new goal in life. ;)
I think the odds are very high.  That, search the city, guild fued, death's presence, and jarard's orders are the only rares in this set.  At least from opening hunderds of packs in draft.  Other people around me keep smiling and saying something about some dual land or other things they open but I'm pretty sure they are liars or just need glasses.



This actually made me laugh out loud. Same thing's happened to me time and time again.

I found Carmen Sandiego before you were born unless you're Zlehtnoba.

Although nothing is more painful than trading away all those copies of a 'jank mythic' like Craterhoof Behemoth only for it to massively spike two weeks later and then you never open any more ever.
Immature College Student (Also a Rules Advisor)
Probability of getting a Firemind's Foresight in a booster:
2/121

Probability of getting two Firemind's Foresight in two consecutive boosters:
2/121 * 2/121 = 4/14641 = 0.0273205%

I do not know what the probability of getting a foil is.
Here's my stab at the calculation.

There are five ways you can get three or more of the same rare or mythic rare in two packs in RTR:

Pack with foil and regular, then pack with regular (and either no foil or a different foil)
Pack with foil and regular, then pack with foil (and a different rare)
Pack with regular (and either no foil or a different foil), then pack with foil and regular
Pack with foil (and a different rare), then pack with foil and regular
Pack with foil and regular, then pack with foil and regular

I have no idea what the actual distribution of foils are.  I don't think they're proportional to the total non-foil cards printed.  Let's assume no distribution by rarity, that there's a 1/274 chance that a foil is any given card from Return to Ravnica

Probability of a foil in a pack: Assume to be 14/67, given one foil per 67 cards and 14 cards per pack.

Assume no pack errors, so 2/121 packs have a specific rare in the rare slot, and 1/121 packs have a specific mythic rare.

Let's take your specific card example, Firemind's Foresight.  We'll calculate the probability that in two packs, you open three or more Firemind's Foresights.  Given the probabilities listed here, the probabilities for each option above are:

(14/67)(1/274)(2/121) x (2/121)(53/67 + 14/67 x 119/121)
(14/67)(1/274)(2/121) x (14/67)(1/274)(119/121)
(2/121)(53/67 + 14/67 x 119/121) x (14/67)(1/274)(2/121)
(14/67)(1/274)(119/121) x (14/67)(1/274)(2/121)
(14/67)(1/274)(2/121) x (14/67)(1/274)(2/121)

The sum of the probabilities of the five options is 4.35 x 10^-7.

For a given mythic, the probabilities are as above, except replace 2/121 with 1/121 and 119/121 with 120/121.  The sum of the probabilities of the five options are 1.14 x 10^-7.

Now add together 53 of the first and 15 of the second to get 2.48 x 10^-5.

^
|
This is similar to how I did it except I didn't include the possibility of getting 4 of the same rare.  Couple issues here I think:
1)  Gatherer only gives 254 cards in the set.
2)  There are 15 cards in a pack (10 common, 3 uncommon, 1 rare/mythic, 1 land/foil).

Also, the distribution of foils is critical since you need atleast one foil to get 3 matching rares.  Until someone can dig up some info on the distribution of rare foils, the results don't bare a whole lot of weight.
Last night I opened two RTR packs, and got:
1 x Foil Firemind's Foresight and 1 x Firemind's Foresight in one pack
and:
1 x Firemind's Foresight in the other pack!

The more I started thinking about it, the more complicated it seemed to be to work out the odds of that happening, so I thought I would pose the question here:

Q: What are the odds of getting 3 matching rares (inc. mythic rares) in two packs?
(not necessarily a particular rare, just rares that match)

Wow!

Well, if the rare isn't a mythic rare, the chances are 8 times higher.

For a particular Mythic Rare, the chance would be 1/121 for the first pack, times 1/121 for the second for the regular rare to be that particular rare.

The chance of a foil... if it's one in 75 cards, as sometimes noted on the booster packs, that would mean one in every six boosters has a foil. The frequency of rare cards is supposed to be slightly higher for the foils, so I don't have an exact figure to work from. If we assume the ratio is just 9:3:1 or (3:1),(3:1),(3:1) (as if there were nine commons in a pack) for the foils for simplicity...

then if there is a foil, the chance that the foil would be a particular Mythic Rare would be 1 in 13 times 121.

So for each of the 15 Mythic Rares, we get a probability of (1/121) * (1/121) * (2 * (1/1573) * (1572/1573))

And for each of the 53 Rares, it would be (2/121) * (2/121) * (2 * (2/1573) * (1571/1573)).

So that works out to ((15*2*1572) + (53*16*1571))/(121*121*1573*1573), or 0.0000380760563328485... .

But I see from the comments above that this was already worked out properly above.

Coming up with weird ideas to make everyone happy since 2008!

 

I have now started a blog as an appropriate place to put my crazy ideas.

Gatherer only gives 254 cards in the set.

There are 249 cards plus 25 basic lands.
Gatherer only gives 254 cards in the set.

There are 249 cards plus 25 basic lands.


I'm going on the record saying I hate details.
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