three times the same card drawn at the start of a match - randomizer issue?

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Comments

  • TheHunter
    TheHunter Posts: 319 Mover and Shaker

    @naabaldan said:
    Sorry guys, even 1% is not true.
    This is quiet easy:
    You have a deck of 40 cards 10 different ones each 4 times.
    The chance to pull one specific card is 1/40. As there are 4 equal cards in that deck, chance is multiplied by 4=4/40 or 1/10. Simple.
    Second step. You drew one card. Now the decks has 39 cards. Our selected one from the first draw is 3 times in the deck. Chance to draw one specific card now is 1/39 and that one and only card that has three left is 3/39 or 1/13.
    Third step. The deck has now 38 cards left. Drawing a specific card now is 1/38. There are still two cards left in our deck we drew the first and second time, so the chance is now 2/38 or 1/19.
    Allover chances are multiplied: 4/40 * 3/39 * 2/28. And we have 10 different cards we can do the same thing with:
    This leads to 4/40 * 3/39 * 2/28 * 10. Simple as it is. Equals 240/59280 or 0,00404858 or about 0,4% of each game we play. In other words: each an every 250th game I start will beginn with three times the same card.
    This is realy simple probability calculation I learnd at school, even though it is like 45 years ago.

    Every day's a school day!

  • jtwood
    jtwood Posts: 1,285 Chairperson of the Boards

    It was not a snide comment at you @Tremayne. It was an observation that we live in the era of alt-facts and anti-science. People would rather believe a comfortable conspiracy theory - that the game cheats - than to believe simple science and mathematics. Nothing you can show or explain will convince them, so the exercise is moot.

  • Tremayne
    Tremayne Posts: 1,673 Chairperson of the Boards

    @jtwood said:
    It was not a snide comment at you @Tremayne. It was an observation that we live in the era of alt-facts and anti-science. People would rather believe a comfortable conspiracy theory - that the game cheats - than to believe simple science and mathematics. Nothing you can show or explain will convince them, so the exercise is moot.

    I didn’t think so, but I wanted to be sure. :)

    And sadly you are correct in your observation. :(

  • Volrak
    Volrak Posts: 732 Critical Contributor
    edited April 2020

    I noticed this thread because I was tagged (thanks @Tremayne), so here are my comments:

    @naabaldan said:
    Drawing the same card three times at the beginning of a match should result into a shuffle of the cards and redraw.

    It's an idea. I agree it's usually a bad player experience to start with 3 of the same card. If that possibility was removed, I don't think anyone would miss it.

    @naabaldan said:
    240/59280 or 0,00404858 or about 0,4% of each game we play.

    Yes, correct. It can be calculated either with the first pick as 100%, as Feden suggested (1 x 3/39 x 2/38), or not, in which case you also have to account for the different cards it can happen with, as naabaldan did (4/40 x 3/39 x 2/38 x 10); both are correct. We know the starting library is 10 copies of each card in the deck; Xucachris's 1% figure would be right if not for that.

    @naabaldan said:
    each an every 250th game I start will beginn with three times the same card.

    Not quite. 1 in 250 is just the average. The difference is technical but important (e.g. because a run of 250 games with multiple triple-draws does not prove non-randomness). The way the numbers work out in this case means that if you take any one set of 250 random games, you'd get the following outcomes with given probabilities:

    • 36% chance of no triple-draws
    • 37% chance of a single triple-draw
    • 19% change of two triple-draws
    • 6.3% change of three
    • 1.6% change of four
    • 0.3% change of five
      And so on with increasingly diminishing odds of getting even more.

    @jtwood said:
    You would also need to provide a count of how many total matches you had played to help with calculating a ratio.

    Yes; that'd be very, very helpful for demonstrating whether there seems to be a problem or not. The total number would help not only by giving a sample ratio, but also by giving a confidence on how well that sample ratio is representative of the true long-term ratio. The value of the screenshots is sadly diminished without recording just as carefully how many games they came from.

    @Tremayne said:
    If someone complains about an issue any evidence is brushed aside

    At least as often I see lack of evidence, or weak evidence, being presented as though it were strong. Both things happening in the same discussion is not unusual. Gathering statistical evidence is hard, which is why usually nobody bothers to do it even if they know how, and also why I'm very interested when people propose things like this:

    @naabaldan said:
    Actually I am at 850+ unopend booster packs. I will record opening them when I reach 1000 packs.

    My suggestion to maximise the value of your data is to record the pack type of each and every pack opened (because drop rates differ per pack); and when a card appears twice or more in a pack, what the pack type of that pack was as well as the rarity (or even better, the full card name) of that card.

  • bk1234
    bk1234 Posts: 2,924 Chairperson of the Boards

    [MOD NOTE] @naabaldan I would like your permission to move this from the Bugs & Technical Issues subforum to the General Discussion subforum. Based on the outcome of this discussion I believe that is a more appropriate place to continue it. Do you agree? [BK]

  • naabaldan
    naabaldan Posts: 552 Critical Contributor

    Even if it feels like a bug, i am ok with moving this. And are you sure, we have an "outcome"? Tell me more ...
    (sarcasm)

  • naabaldan
    naabaldan Posts: 552 Critical Contributor

    Yes, correct. It can be calculated either with the first pick as 100%, as Feden suggested (1 x 3/39 x 2/38), or not, in which case you also have to account for the different cards it can happen with, as naabaldan did (4/40 x 3/39 x 2/38 x 10); both are correct. We know the starting library is 10 copies of each card in the deck; Xucachris's 1% figure would be right if not for that.

    4 copies pls. 1% is still wrong. And to draw a specific cards chance is still 1 out of forty, never 1/1. 1/1 means you pull one card out of a stock of one card. Misleading is, you pull a card, that 100% sure, but what card is it? you have 40 different cards, with 4 times each card has the same image. In case you decide to destinct those cards the chance is 1/40, otherwise you can not determin which of the 4 cards you drew its 4/40. Still.

    I hope no mathematician will ever follow or read this thread.

  • jtwood
    jtwood Posts: 1,285 Chairperson of the Boards

    @naabaldan said:

    I hope no mathematician will ever follow or read this thread.

    I think this is the funniest thing I've ever seen on these boards :smiley:

  • Volrak
    Volrak Posts: 732 Critical Contributor

    @naabaldan said:
    4 copies pls. 1% is still wrong.

    As I said. It would be right if not for the way the library works.

    @naabaldan said:
    And to draw a specific cards chance is still 1 out of forty

    Again as I said. As I pointed out, there is also an alternative (and simpler) way to arrive at the same (correct) answer which does not require a 1/40 factor anywhere in the calculation.

  • bk1234
    bk1234 Posts: 2,924 Chairperson of the Boards

    [MOD NOTE] Moved with OP’s permission (Bk)

  • Larz70
    Larz70 Posts: 137 Tile Toppler
         @naabaldan I came across your post and was impressed by the screenshot.  Math is hard so I can totally relate yo the different calculations that came up.  It's 5%, no it's 1%, no it's actually .4%.  After thinking about it some, I'll go with .4%.

        As Volrak pointed out, that's not all there is to it.  If you play 250 matches, it's not 100% guaranteed that you'll come across 3 same cards in a row once.  You might see it zero, one, two or more times.  In your case, I think you mentioned here that you saw 6 in a span of two weeks playing about 200 matches.  That feels, to me, like a very high number, 6 in 200, when we already agreed that you should see about 1 in 250.  But is it impossible?

         I ran a quick Monte Carlo simulation where I had 1,000 players play 200 matches and counted how many games they played where the same first 3-cards showed up in a row.

    Here are the results:

      0       1       2        3        4      5 
    438   373   139     43       6      1 

         I don't know how to format the table above but basically, 438 players did not see the same card in their first 3, 373 saw one instance, 129 saw 2, 43 saw 3, 6 saw 4, and 1 player experienced drawing the same card as their first 3 5 times in 200 games.  No one saw 6 or more.

         When I expanded to 2,000 players, 

      0         1         2         3       4        5 
    902     710     296      71     17       4 

         Still no one saw it 6 times.

         At 5,000 players
       0          1      2      3      4     5    6 
    2236   1785  739  201   35    3    1  <--  (woot, I finally found naabaldan!  :D  )

         I finally got 1 player who saw it 6 times. 

         I also tried 250 games with 1,000 players and I got. 
      0      1      2      3     4     5    6 
    364  371  178   63   19    3    2 <-- (naabaldan, you are not alone ;P )

         2 players experienced drawing the same first three cards 6 times out of 250 games.

         TLDR: I don't know what the exact percentage is (Volrak stopped at <0.3 will see it 5 times out of 250), but if I ran the simulation long enough, it should converge to a pretty good estimate of the probability.  Based on the few sim runs I tried, it seems that in a population of players playing 200 matches for two weeks, somewhere around 1 in 5,000 of them will see the same first 3 cards 6 times.  Playing 250 matches however, around 1 in 500 will see it 6 times.  (Remember, that the sim is limited to number of runs so these are all estimates.  I can run it for longer but I don't really want to wait that long to get the results.  The fact that the sim found a player or two experiencing what naabaldan experienced means that what happened was random chance and within the confines of what's probable)

      

  • naabaldan
    naabaldan Posts: 552 Critical Contributor
    @Larz70 thank you for your efforts. I know that everything can happen.
    To fill the list: within the last coalition event this happened 2 times again.
    What i do not understand is, no one reports: it happens to me too...
    So I feel alone with this issue. I am the only one.
  • jtwood
    jtwood Posts: 1,285 Chairperson of the Boards
    naabaldan said:
    @Larz70 thank you for your efforts. I know that everything can happen.
    To fill the list: within the last coalition event this happened 2 times again.
    What i do not understand is, no one reports: it happens to me too...
    So I feel alone with this issue. I am the only one.
    Considering the wide array of other bugs, and the necessary mathematical understanding required to realize something is amiss, I’m not surprised people have not been focused on this topic. 
  • madwren
    madwren Posts: 2,259 Chairperson of the Boards
    Drawing 4 of the same card in my first four cards has happened to me multiple times as well. You're not alone, but many people who have this occur either 1) understand it's happenstance, or 2) aren't on the forums. We're a small subset of the playerbase. 
  • Barnabes
    Barnabes Posts: 76 Match Maker
    naabaldan said:
     
    What i do not understand is, no one reports: it happens to me too...

    Because when it happens, I am too busy evaluating how to win the game with what I have instead of thinking that the computer is trying to screw me over. Three copies in hand of Scapeshift with Sarkhan Unbroken usually means I get to ultimate next turn. Three copies of Hazoret's Undying Fury usually means I win if I can cast one. Three copies of Nicol Bolas means I can cast one and it will help me draw to my other cards, negating the poor start.

    I would have to be repeatedly drawing three identical bad/mediocre/situational cards (say a cheap support removal spell) out of my deck before it would even register as a potential issue. And I have to say, it has not happend to me yet.

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