Two times the same card in a row when opening booster packs

naabaldan

This is something everybody experienced when opening boosters: the same cards is pulled twice in a row.

Nalthazar also noticed in one of his videos when he opens packs, that this happened, often.

To go into deep i saved every booster pack for a long time, until i reached 1000 packs and than start opening them recording the cards pulled and than counting the occurences of pulling twice the same card in row or in a pack of 5 cards at all.

The results are astonishing: every 10th of an opened booster pulled twice the same card in a row. Yes its 10%

Once i even pulled three times the same card in a row and even free three card boosters can contain twice the same card in row.

Intrestingly to notice: there was never the same card in a booster not pulled in a row. And there is another strange effect: this can happen to any rarety. I did pull commons, uncommons, rares, mystics and even a masterpiece twice in a row.

First of all: pulling a duplicate within one 3 or 5 card booster should never happen, not by error and not by chance. It is very easy to prevent this. But the reason why I decided to post this issue here is: I suspect an unwanted programming effect.

If it is not an error: how can it be that 10% of every opened booster contains the same card twice in a row but NEVER in any other combination like the first and the last card?

I don't want to discuss how randomness is programmed or how to select a card from a set randomly by generating random numbers. I want to know, how often this happens to you!

And finaly I want this to be fixed, so it could not happen any more, because we are losing a chance of obtaining a new card we where chaising for when this happens.

Tremayne

I have seen this many times but I have no cooperative data.

Nice work

Volrak

Back-to-back dupes almost never happens for me.  I'd be very interested to see your data @naabaldan if it's available somewhere.

naabaldan

The pack opening was set wise, everything else is not efficient enough to do in time.

I opened following sets in that order, first number Boosters, second number 2 dupes in a row (one time three)

• BFZ 10 / 1
• SOI 14 / 2
• HOD 12 / 1
• IXL 28 / 2
• RIX 24 / 2
• Core2019 28 / 4
• GRN 71 / 7
• RNA 69 / 6
• WAR 109 / 11

More to come soon, when i checked the remaining videos/sets..

BTW: Opening BFZ and SOI boosters took much longer than boosters from newer sets.

I will setup a video combining every dupe in a row and ask Nalthazar to post it on his youtube site.

Bubbles_CS

I open all my packs and I do see the same card twice in a row in a pack. I have also experienced three-in-a-row.

@naabaldan do your data reveal randomness across pack position? I remember in years past finding that rares and mythics usually showed up in the last pack position, sometimes in the first and rarely in another position. Although more recently I have observed a more even distribution throughout the pack, if there was some sign of disparity in position it could indicate that individual pack positions are coded in non-identical ways.

also, did you find non-sequential dups in the same pack, and do you or anyone with an understanding of statistics know how often random chance should drop two in a row?

naabaldan

"did you find non-sequential dups in the same pack"

--> NEVER, the only dupes are coming in a row

that is the reason why i suspect an issue in the coding of the randomizer or the code fragment that uses the randomizer to select a random card from a set of cards.

"How often random chance should drop two in a row"

--> I was avoiding this, my last post regarding drawing the same three cards in a row at the beginning of match was a messup of answers (not all of course)

But, you asks for it - I see what is comming now

To find out how often this can happen, we have to ignore for simplicity rares, mystics and masterpiece in the first place and assuming commons and uncommons have equal chance (what is not true, but close enough).

So drawing one specific card has (number of commons + number of uncommons) or (60+65) or 125 different possibilities for a common booster pack. Numbers differ throughout different sets, but the calculation is the same, not counting odyssey and torment boosters because of there completely different chances.

The second draw has the same chance to draw a specific card, and chances are multiplied. To calculate the chance of one special card drawing twice in a row - here are the numbers: 125*125 or 15625 different combinations are possible but only 125 are 2 in row, so chance is 1/15500 or 0.00645%. This is true for 2 draws, but we are drawing 5 times, so we have a combination of draws 1st 2nd, 2nd 3rd, 3rd 4th or 4th 5th, saying this can happen in 4 different ways = 0.0258%, ignoring the pairing we have 10 different ways or 0.0645%. in other words: for pulling a pair of the same card in a row for ONE TIME you need to open 3875 packs (statistically) or 1550 packs in case you are ignoring pairing.

Ironicaly even though the chances to get a pair within 5 draws in any positon are higher it never occurred within all packs I opened.

The calculation is simplified and ignores the different chance of drawing commons, uncommons, etc. but it is close enough to get a feeling that 10% in reality is way to often compared to 0.0258%.

I apologyze for my bad use of english here, this can be expressed in much better words (it's not my native language). 

BlessedOne

@naabaldan
I really appreciate truth expressed in math. Thanks for nerding out.

Volrak

naabaldan said:

So drawing one specific card has (number of commons + number of uncommons) or (60+65) or 125 different possibilities for a common booster pack. Numbers differ throughout different sets, but the calculation is the same, not counting odyssey and torment boosters because of there completely different chances.

The second draw has the same chance to draw a specific card, and chances are multiplied. To calculate the chance of one special card drawing twice in a row - here are the numbers: 125*125 or 15625 different combinations are possible but only 125 are 2 in row, so chance is 1/15500 or 0.00645%.

Your simplifying assumptions are quite reasonable, but using those, the chance of the next card matching the previous card is not 1/15500, but 1/125.  That result can be understood in several ways:

• The first card may be any card; the second card just has to be one specific card out of the 125 possibilities (i.e. the same card as the first card), so the chance of a matching sequential pair is 1/125.

• There are 125*125 combinations of two sequential cards; 125 of those are matching pairs; therefore the proportion of matching pairs from all the combinations (and thus the chance of a matching sequential pair) is 125 / (125*125) = 1/125.

As you say, you can then multiply by 4 to get the expected chance for a sequential pair in a 5-card pack: 4 * 1/125 = 3.2%  But that's still quite a bit lower than your observed 10% figure.  Another key unexplained observation is seeing dupes within a pack only ever back-to-back, and not in any other arrangement.  If each card is independently random, the chance of seeing a dupe somewhere in the pack which isn't back-to-back would be around 5% - higher than the chance of seeing it back-to-back - but you're seeing it 0% of the time.

It seems like there are two different phenomena requiring explanations, but there is at least one theory in which one observation actually serves to explain the other:  Imagine if pack contents were determined randomly, and then sorted by some method which results in dupes within a pack always being back to back.  What we'd expect to see in this scenario (using the 1/125 simplification) is roughly an 8% chance of back-to-back dupes in a given pack, and a 0% chance of dupes anywhere else in the pack.  That is starting to sound very much like what you have observed.  If the theory is correct, then there should be some consistent ordering to cards within a pack, which may be deducible.

bk1234

Fairly recently I pulled Zetalpa 3 separate times from  5 card packs in a week. I just thanked RNGesus for my crafting orbs and went about my week. He’s like that. I’ve accepted it. This isn’t the first time and won’t be the last. 

But I wanted to drop by and thank you both for the stats and maths. When in doubt I err on the side of math. It generally doesn’t lie, unless it sets out to. 

naabaldan

maybe i was wrong with that 125x125 - 125 and chances are higher, i will review this.

Bubbles_CS

Fantastic stuff, thanks so much @naabaldan and @Volrak! Sorting would definitely explain why only this type of dup shows up and if it is true that rares and mythics appeared mostly last, sorting might explain that too and might even just be an old piece of code that only has side effects now that it doesn’t move the rares.

I am almost afraid to ask, but I will anyway If we assume sorting is a thing, that means we would observe an adjacent dup no matter where in the pack the two dup Cards appear while the pack is being created. Does that get us closer to 10%?

Volrak

Bubbles_CS said:

If we assume sorting is a thing, that means we would observe an adjacent dup no matter where in the pack the two dup Cards appear while the pack is being created. Does that get us closer to 10%?

Yes (that's the 8% in my bottom paragraph)

Bubbles_CS

Volrak said:

Bubbles_CS said:

If we assume sorting is a thing, that means we would observe an adjacent dup no matter where in the pack the two dup Cards appear while the pack is being created. Does that get us closer to 10%?

Yes (that's the 8% in my bottom paragraph)

Whoops obviously I missed that. Thanks for pointing it out, @volrak!