More fun with stats and duplicates

julianus

Though I am not a statistician, I was so entertained with the discussion in this thread on Masterpiece duplicates (http://forums.d3go.com/discussion/61875/masterpiece-collection-duplicates#latest) that I was inspired to do a bit of testing of my collection of SOI rares.

Context: I'm not a programmer, so I didn't write any code to run a simulation, but I did set up a spreadsheet to do multiple simulated random draws.

Using Octal9's site, I saw that there are of 59 rare cards in Shadows and Eldritch Moon. I have 28 total rare cards from those expansions, including duplicates, so I simulated 28 random card draws out of 59 total cards, and calculated the mean and standard deviation. I did 100 total draws, grouped into 10 sets each.

The results:

• Average number of individual cards out of 28 draws: 22.4
• Standard Deviation: 1.58
• Number of  individual rare cards in my collection: 18
• Number of individual rare cards drawn at 95% Confidence: 19.24 to 25.56
• Number of individual rare cards drawn at 99.7% Confidence: 17.66 to 27.14

The confidence intervals are, of course, assuming a normal distribution.

julianus

Addendum:

I just checked my inbox in game to see the rare card I received from the latest Quick Battle. It was from Zendikar, a third copy of one of the 14 rare cards I have from that set...

Volrak

Nice job - I get the same results.  But some assumptions are worth checking:

• Were you already playing mtgpq before EMN was available?  If so then not all of your drops would have been pulled from the SOI+EMN rare pool (59 cards); some significant subset might be from the SOI-only rare pool (35 cards).
• Similarly, if you won any event prizes which were "EMN rare" or "SOI rare" (as opposed to "EMN or SOI rare"), or if any SOI/EMN rares were exclusive at any point, it should also be accounted for.

(It's things like this which make Origins so much easier to analyse!)

julianus

Volrak said:

Nice job - I get the same results.  But some assumptions are worth checking:

• Were you already playing mtgpq before EMN was available?  If so then not all of your drops would have been pulled from the SOI+EMN rare pool (59 cards); some significant subset might be from the SOI-only rare pool (35 cards).
• Similarly, if you won any event prizes which were "EMN rare" or "SOI rare" (as opposed to "EMN or SOI rare"), or if any SOI/EMN rares were exclusive at any point, it should also be accounted for.

(It's things like this which make Origins so much easier to analyse!)

I haven't been playing that long, and EMN was definitely available when I started. I'm pretty sure the current pack was already Kaladesh when I joined, though I wasn't entirely clued into all of the different packs at first. I know that I didn't win any SOI or EMN set-specific prizes from events.

I did another quick stat calculation based on the above numbers, to find the probability that I would get 18 or fewer individual rare cards out of 28 total. And, again assuming that it's a random selection of cards following a normal distribution, that possibility is less than 1%.

So, while I can't completely rule out the chance that I just got bad luck when drawing my rare SOI/EMN cards, if it's genuinely random then my luck was very bad indeed.

Volrak

julianus said:

I haven't been playing that long, and EMN was definitely available when I started. I'm pretty sure the current pack was already Kaladesh when I joined, though I wasn't entirely clued into all of the different packs at first. I know that I didn't win any SOI or EMN set-specific prizes from events.

I did another quick stat calculation based on the above numbers, to find the probability that I would get 18 or fewer individual rare cards out of 28 total. And, again assuming that it's a random selection of cards following a normal distribution, that possibility is less than 1%.

So, while I can't completely rule out the chance that I just got bad luck when drawing my rare SOI/EMN cards, if it's genuinely random then my luck was very bad indeed.

Indeed.  I get the odds of 18 or fewer non-dupes out of 28 random drops from 59 choices as being 1.3% (1 in 77), which more or less agrees with your figure.

Edit - Added the histogram (from 10 million runs)

 

julianus

Volrak said:

Indeed.  I get the odds of 18 or fewer non-dupes out of 28 random drops from 59 choices as being 1.3% (1 in 77), which more or less agrees with your figure.

Edit - Added the histogram (from 10 million runs)

 

Fancy.

I couldn't do anything like 10 million calculations in Excel, so I'm happy to accept your more accurate estimate. As you note, it certainly doesn't change the interpretation that I was very unlucky with my rares.

Either that or, you know, the other interpretation.

Scotcamp

Someone riddle me this for stats.

I have only 1 of the 8 elite pack cards, so I gamble that with a 7/8 chance of getting something new.   
What did I get...the only mythic I had before. 
Refunded from D3.  Open the pack....what did I get....yep the same mythic.
Ask D3 for compensation...Got it....

Drumroll please...What did I get?????  
You got it...same mythic card.....anyone need a lizard?

Ekkias

@Scotcamp I'm just going to nitpick for a second, because we're in a thread about statistics...

The cards in Elite Packs are definitely weighed, so it's not correct to say you had a 7/8 chance (87.5) for a non-dupe.

There's probably only a 3 - 5% chance for a MP draw, with the other 95 - 97% being a mythic. Assuming the mythics are equally weighted (I highly doubt they are), thas actually a 80.6 to 81% chance of a non-dupe.

Either way, under that assumption drawing that 1 dupe 3 times in a row is a less than 1% chance.

Obviously the mythics are not equally weighted and/or their pseudo-RNG code is skewed toward drawing dupes.

Scotcamp

Ekkias said:

@Scotcamp I'm just going to nitpick for a second, because we're in a thread about statistics...

The cards in Elite Packs are definitely weighed, so it's not correct to say you had a 7/8 chance (87.5) for a non-dupe.

There's probably only a 3 - 5% chance for a MP draw, with the other 95 - 97% being a mythic. Assuming the mythics are equally weighted (I highly doubt they are), thas actually a 80.6 to 81% chance of a non-dupe.

Good point.  Let me be perfectly clear.  I would be happy with just a mythic I don't have.  I have no expectation of a MP.  I wanted one of the 7 cards that I didn't own. that was my only expectation.