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A Note on Probability

by on March 20, 2014

Gandalf2

When designing and talking about decks, players will often use words such as “consistency” and “reliability” to discuss how often certain cards will emerge over the course of many games (not just one game). Oftentimes, these discussions will be linked to the size of one’s deck and the number of copies included of a given card. Related to all this is the reality that there will often be one particular card that you are dying to see in your opening hand, whether it’s that Steward of Gondor to get your resource engine running, Light of Valinor to power the Glorfindel machine, or Gondorian Shield to turn Beregond into a one-man wall. Still, I’ve often wondered what the actual probability is of drawing such a card, and how it’s affected in real terms by deck size and number of copies, beyond just the common kernels of wisdom traded between gamers. This has real applications, as it can help to guide deck building decisions.

Disclaimer: I am not a mathematician, nor do I claim excessive math expertise. No numbers were harmed in the making of this article.

If you have an unhealthy aversion to math, you may want to look away, as here there be numbers. In order to calculate the probability of drawing a desired card, I’m first going to calculate the probability of NOT drawing said card in my opening hand, as well as the card drawn in the first turn (7 cards total). Why? Because math said so. I’ll assume the ideal scenario of 50 cards in the deck and 3 copies of the card I want. Let’s just say Light of Valinor because Glorfindel would want it that way.

This is what happened to the last guy who didn't fetch his favorite attachment...

This is what happened to the last guy who didn’t fetch his favorite attachment…

47/50 is the probability of not drawing Light of Valinor as my first card, 46/49 is the probability of not drawing it as the second card, etc. Multiplying these probabilities will give me the odds of all of these events occurring, meaning that I don’t draw Light of Valinor at all. Thus,

prob

Of course, remember that 63% is the chance of NOT drawing the desired card. If we subtract this percentage from 100, we get 37%. Thus, there is a 37% chance of drawing Light of Valinor in your first 7 cards. Those aren’t great odds. However, we can’t forget about our old friend, the mulligan, which just might save Glorfindel’s day…and my own skin. However, when considering the mulligan, we have to consider that we would only see the first six cards of the opening hand before making the decision to mulligan, which is only a probability of 32%.

The odds of drawing Light of Valinor for the mulligan, when you would see all 7 cards (since you can’t mulligan again) would be the original 37%, and we must combine these probabilities together to figure out the overall odds over the course of two tries. What we get is a 43% chance of not drawing Light of Valinor in our opening hand AND the mulligan hand. This means that overall we have a sizable 57% chance of drawing a desired card with 3 copies in a 50 card deck.

That’s pretty decent actually, and much more likely than we might think if we just consider there being 3 copies of a card in a huge 50 card deck. This means that for every 5 games we play with a given deck, we should draw the card we want in every 3 games.

However, now let’s examine a few different variations on this scenario to see how deck building affects consistency and thus our success in the game. How would our probability of drawing Light of Valinor change if we only included 2 copies instead of 3? After all, those additional copies after the first are pretty useless. Surely we can afford to shed one copy? Not if you don’t want an angry Glorfindel on your hands, as the probability actually drops all the way down to 43% if you only include 2 copies. This wouldn’t be too bad if you’re talking about a card that is nice to have, but not necessarily essential. However, a 43% chance means that you’d be drawing a key card in less than half of games, which is not a model of consistency. Therefore, you should always include 3 copies of any card that you want to draw in your opening hand, even if it is unique.

I'm here to release birds and murder those who fail me, and I'm all out of birds.

I’m here to release birds and murder those who fail me……and I’m all out of birds.

Let’s take another scenario. Many players ask whether keeping to the 50 card minimum is really that important. How does increasing the deck size affect the odds of drawing a key card? What if we increased our decks to a massive 60 cards to include all those cool effects we normally have to cut out? The probability would fall to 50%, which actually isn’t as precipitous a decline as I was expecting. Thus, the number of copies of a card is actually a bigger factor in the probability of drawing it than the deck size (assuming you’re not going crazy and inflating your deck to ungodly sizes). In fact, you’d have to go all the way up to around a 70 card deck before you would equal the impact of removing just one copy of the desired card. Of course, keep in mind that consistency means more than simply the probability of drawing a certain card in your hand, so don’t take this as an endorsement for building 60-card decks. 50 card decks are still the ideal if you want maximum consistency for your deck in general, not just in terms of one card (it would take further statistical modeling to prove this, which is beyond my attention span at the moment, but you’ll have to take my word for it). However, if you’re worried that using a 52-card deck (the size of a deck of cards for standard games like poker) will impact your chances of drawing a certain card dramatically, the actual impact is negligible (a 1% decrease).

Finally, how does including card draw or “fetch” cards improve the odds of drawing a needed card in the first turn? Let’s take a look at our Hobbit friend, Bilbo Baggins, who could actually allow you to start off with 8 cards instead of 7, assuming you’re the first player. Bilbo would increase your chances to 60%, which is not a huge difference, but not inconsequential. What about including 3 copies of a fetcher like Master of the Forge (you’re chances of drawing at least one of either the attachment you want or the Master is really high, around 83%). Assuming you drew and were able to put into play a copy of Master of the Forge, how would that affect the probability of getting access to Light of Valinor on the first turn? It increases to a magnificent 71%!  What about a popular card draw effect like Daeron’s Runes, which would allow you to grab 2 extra cards for no cost? This would up the probability to 63%. The lesson here is that if you are looking to grab a particular card on your first turn, a “fetch” ability that can look at more cards than a generic card draw ability is the best choice if you have to pick between them. Again, remember that getting one particular card is not the key to overall consistency or success necessarily, so don’t take this as meaning that Master of the Forge is better than Daeron’s Runes in a general sense, but this example does give an insight into how statistics and probability can illuminate deck building decisions for particular situations.

Hopefully this short segment has not bored you to tears but instead provided a tool for deck design, specifically for those decks that center around one particular card. Perhaps you’ll feel better informed now the next time you consider the wisdom of relying on drawing Resourceful during the first round for a deck that starts off at 20 threat. There definitely is a whole world of further statistical analysis out there, and perhaps some intrepid souls will continue this exploration. For now, I’ll end with a handy chart for the hardcore and the delightfully nerdy.

probchar

 

* Reader GeckoTH pointed out that you wouldn’t actually see the seventh card for your first  before you make the decision to mulligan, so I’ve adjusted the math in this article accordingly.

 

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63 Comments
  1. mtpereira permalink

    I did not had the impression that it was so likely to draw a certain card when doing mulligan. Thank you for doing the math for us! 🙂

  2. Great article. On a related note, I think this also highlights the importance of having a method of converting extra unique cards into something of value. Obviously we have Eowyn and Protector of Lorien.

    I’ve been experimenting with the Trollshaw Scout, who tends not to get killed (v. Watcher of the Bruinen) and who can be equipped with a Rivendell Blade to drop multiple enemies’ defenses in one round.

    Any way you can calculate the odds of drawing two particular cards in your starting hand? That math is far beyond my abilities, but I’d be curious to know, for those of us who rely on first turn combos.

    • TalesfromtheCards permalink

      The math gets pretty complicated once you delve into the odds of finding more than one type of card, as you have to start pulling out combinations. However, I did the calculations of drawing both a Rivendell Blade and Trollshaw Scout in your opening 7 cards, and the odds are only around 12%. However, with a mulligan, you have about a 23% chance. So relying on a combo showing up consistently is a bit more dicey it seems.

      • Thanks, Ian. I imagined the odds were pretty low, but thanks for confirming.

  3. Dude, well done! Being a former poker player, I enjoy looking at the odds. The bit about the 1% difference in finding cards in a 52 count deck is a huge revelation, as I’ve been pretty fanatical at keeping decks at an even 50.

    • TalesfromtheCards permalink

      Thanks! I didn’t know you played poker before, or maybe you mentioned it and I forgot. Anyway, I think sometimes I roll with the 52 cards just because it’s standard playing card deck size.

      • A few years ago I was playing Hold ‘Em at least once a week. Started to fizzle out, though, but I do miss it.

        Nerd stuff tends to take priority.

        • TalesfromtheCards permalink

          Well,. Doomtown does have a touch of poker… 😉

  4. Fantastic article!! Really helpful.
    One question though… You mention that you’d have a 37% chance to draw Light of Valinor in the first 7 cards, and if you take a mulligan, you’d have a 37% once again. However, you’d only be able to see the first 6 cards before deciding to mulligan or not. So in your first draw, I think you’d have around a 32.5% chance to see the card, then a 37% chance the second time around (this time including the Turn 1 draw). So, you should have a 57.5% chance to draw Light of Valinor if you mulligan. (I’d need someone to double-check my maths for me though, lol)

    I appreciate you showing how increasing your deck size by a bit may not have as drastic effect as some people like to think.

    • 57.5 is correct.

    • TalesfromtheCards permalink

      That’s a fantastic point, GeckoTH. I completely omitted the decision-making aspect of taking the mulligan in order to make my life (and math) easier, but it’s definitely important to keep in mind. I did the calculations and you’re correct that you only have around 32% odds of seeing LIght of Valinor in your first six cards, before you have to make the decision to mulligan. So around 57% (57.5) is correct for your overall chance. Thanks for the math/logic check!

    • TalesfromtheCards permalink

      Ok, I’ve adjusted the article to take the more accurate approach. Thanks again!

  5. First of all, I think my favorite part of this article is that the end table is purple and green (my favorite colors).

    I think my next favorite part is that you eases my slight anxiousness to be mostly non-anxious with regards to 52 card decks. Most of the time I use 52-card decks and wonder if moving down to 50 would be worth it. Most of the time, I don’t think it is. I think you also freed me up to consider up to 55-card decks.

    Finally I’m wondering about a few factors:
    1) How did you calculate Bilbo’s probability? Did you calculate the probability of drawing with 8 cards and then square it? It seems like you did, but this would not be the probability since if it does not show up in the first hand, that hand will only draw 7, and THEN you’d draw 8 after the mulligan.
    2) It’s interesting (and brain-breaking) to think about the fact that if you DO draw the card and you draw it early, then the chances drop because there are only 2 copies of the card left in the deck, but then again the chances are 100% because it’s already in your hand, so you don’t give a care about the other 2 copies. Probabilities are stupid 😛

    • Oh yea… and what GeckoTH said… because I was forgetting that you actually only draw 6 cards and then the 7th is the draw for the turn. Thanks GeckoTH for knocking some sense back into me. This goes along with my comment about Bilbo.

    • TalesfromtheCards permalink

      Yes, good point. I definitely made some simplifications, which aren’t the best when it comes to math. The adjusted probability would be about a 60% chance given 6 cards for first try and 8 for the second.

  6. jamesarthurharrison permalink

    Something to remember is that while you overall have a 57% chance of hitting that Light of Valinor, is you Mulligan you still only have a 37% chance of hitting the desired card. What does this mean? Basically even if you are missing the card if your original hand is excellent (say the aforementioned Daeron’s runes and a Master of the Forge) you should strongly consider keeping it.

    • TalesfromtheCards permalink

      That’s a good point. I formulated these probabilities as a guide to deck building decisions and strategy (how much can I rely on a certain card showing up given certain factors?), but these numbers can also be helpful to guide the mulligan decision as well, as you point out. If I had a good hand, I don’t think I would play those odds with the mulligan.

  7. catastrophic09 permalink

    Because your opening hand and mulligan hand are entirely separate from one another and both start from a freshly shuffled deck why would the probability change with a mulligan?? Is you chance of getting that one card not always 32% because the mulligan does not change the probability since the deck is fully shuffled each time? For example, think about it, you shuffle your deck and you have a 32% chance of seeing a card, lets say you don’t get it so you mulligan shuffling your deck and now you have a 32% change of seeing a card, etc… I don’t think the mulligan changes your chances, does it?

    • TalesfromtheCards permalink

      It’s definitely not the most intuitive thing in the world, but the probability of two independent events (such as drawing twice from a reshuffled deck) is found by multiplying the two probabilities together. Think about this way. Say I’m playing a board game where I need to roll a 6 on one die to win. Each time I roll that die, the odds are exactly the same (1/6). However, my odds overall of rolling that 6 increase the more re-rolls I have. So if I have 5 chances to roll the die, my odds of hitting a 6 are much better overall than rolling it just once, which does make intuitive sense.

      Here’s some probability stuff (See experiment 3):

      http://www.mathgoodies.com/lessons/vol6/independent_events.html

  8. catastrophic09 permalink

    Ahh.. ok ok now thats starting to make a little more sense. So because you are specifically wanting a certain card, you would continue to mulligan or draw a new hand and each time you do this it increases that overall chance of getting it. At first it is certainly confusing because it seems your chances would be the same but if we had 5 mulligan’s for this LOTR game then we would probably always get that one card we need and stop drawing a new hand once we got it.
    Thanks for the clarification! 🙂

  9. GrandSpleen permalink

    Oh and don’t forget the great equalizer variable of “how you shuffle.” Pretty sure my wife has at least an 80% chance of drawing 4 of the same cards that were in her original hand after she mulligans.

    • Landroval permalink

      I suffer from this! what is the best way to shuffle quick and effectively?

      I’m not joking!

      • When I studied statistics, one of my fellow students had this as their final exam. He found that the statistically perfect way of shuffling a deck of cards was: Seven imperfect shuffles followed by a completely random cut. Each “shuffle” above was conducted by dividing the deck into two imperfect halves, inserting them into each other once, always making sure that the bottom card and the top card was replaced after each shuffle like this:

        1) IIIIIIIIII (the whole deck seen from the side)
        2) IIII IIIIII (divided into two imperfect halves)

        3)
        I I II
        IIIIII

        As you can see, the previous bottom card was furthest to the left originally, and remained the leftmost card of the four card group. Meanwhile, the rightmost card originally was the top card of the deck, and stayed farthest to the right in the six card group. Now you put the top row in the “picture” straight down, so that they sort of filter in among the bottom row, and as you can see, the top and bottom cards arena are replaced. Do this seven times, and then finish with a cut somewhere (you could cut only one card, now and then; it’s still considered a cut of the deck).

        • …and the removal of a line-opening space ruined my “fantastic” show. Try to figure out how the final diagram would look if you move the top row slightly to the right. That’s how I intended to draw it…

    • TalesfromtheCards permalink

      Haha! Only too true. The perfect world of math shatters upon the rocky shores of real world shuffling.

      Landroval, I am no shuffling expert, but you might find it helpful to look up some shuffling videos on youtube, as I’ve seen a few that demonstrate the best shuffling techniques for randomizing your cards (and keeping them safe).

      • Tracker1 permalink

        I started to take shuffling real serious when I took a mulligan shuffled my deck, cut it and proceeded to draw the same six cards. Yikes.

        • TalesfromtheCards permalink

          Ouch! I feel like my shuffling has gotten better simply through playing this game so much. As for OCTGN, it should be nice and randomized, but there’s been times where I swear it needs help learning to shuffle as well…

  10. I can’t believe you posted this. I spent yesterday morning trying to solve exactly this problem and couldn’t. Thank you sir.

  11. Those 1% add up. In a 16 round event like a big magic tournament it will be about 16% of something going awry. I stick to 50-52 cards but the danger is if you start thinking one more won’t hurt you end up with a boated 60 carder.

    • Good point, Jonathan: versus very difficult quests, you want to have a very consistent deck. Against easier quests, you can afford to be a little more sloppy in your deck building, with 1-2 copies of “neat, but not necessary” cards, or a more bloated deck with more options that you see less consistently.

    • TalesfromtheCards permalink

      I tend to use 52 cards as my upper limit. I think it’s important to keep in mind that while a 55 card deck (or higher) won’t hose your chances of drawing one particular card as much as might be expected, it will hurt the overall consistency of your deck.

    • Actually, you’ll have a 16 % chance of something going awry in ONE of your 16 games. Of course, playing more games will increase the chances of something going wrong eventually, but if you increase the chance of something bad happening by 1 % in av game, that means that it will probably happen once every 100 game, as 1 % means “one out of a hundred”.

  12. Tracker1 permalink

    Now I know why master of the forge is one of my favorite and most used allies.
    Thanks!

    • TalesfromtheCards permalink

      He only gets better and better. Such a good ally and well worth the 2 resources, especially since the most powerful and important cards tend to be attachments.

  13. This is a great post, but it really makes me question some of FFG’s decisions on the core set. Having only one copy of a really powerful attachment like Unexpected Courage or only two copies of Steward of Gondor really messes with beginner deck building.

    • TalesfromtheCards permalink

      Agreed. It has surely been one of the most debated aspects of the Core Set. FFG says that it does this for Core Sets so that they can include a greater quantity of different cards, but you do have to wonder why include only 1 Unexpected Courage instead of 1 copy of a weaker card?

  14. DC06675 permalink

    This article was really a revelation! There have been a number of times where I went from x3 to x2 and even x1 of cards I really like (even if they aren’t the most powerful cards) simply so I could get to the 50 card limit because I read everyone saying that 50-card decks are the most efficient. While they still may be more efficient I realize I can put in a few more ‘fun cards’ in my deck!

    • Yes, you could probably think about it like this: If you include two copies of your favorite card, and your deck size is a slim 50 cards, then that card fills only 4 % (or every 25th card) of your deck. If you include a third copy (without removing anything), you’ll have 51 cards in your deck, and your favorite card suddenly fills 5.9 % (or every 17th card) of your deck. Actually, putting in as many as 74 cards in your deck, including three copies of your favorite, still gives you a better chance to pull it, than with the original 2 of 50.

  15. Mithrandir1119 permalink

    Wow, I had just posted basically this same thing on BGG a week or so before your post
    http://boardgamegeek.com/filepage/101217/probability-of-drawing-a-card

    Good writeup

  16. Ian, your point about the power of “fetch” cards is important here. The Seeing-Stone is a great example because you can search your entire deck for a Doomed card. I’ve been using 3x of it in my Doomed decks, along with 3x Legacy of Númenor, and it almost guarantees I’ll have six resources on Turn 1 instead of three.

    • TalesfromtheCards permalink

      Definitely. Exploiting The Seeing-stone and Legacy of Numenor is my next project 🙂

  17. eucatastrophe permalink

    This article was so helpful! Thank you! Have you ever thought about a full table, with number of cards in the deck as the columns, and number of cards in the deck as the rows? Often I wonder how the odds will change in having, say, 6 or 9 of the same *type* of card (i.e. threat reduction) in a deck. Could be cool to see all the possibilities.

    • TalesfromtheCards permalink

      Glad you enjoyed it! I think there’s definitely much more that could be done in this realm, and your idea of a full table is a great idea. I hope to add another article along these lines in the near future.

      • eucatastrophe permalink

        That would be awesome. I’ve come back to reference this article many times.

      • eucatastrophe permalink

        Question: How did you combine the first set of probabilities (32%) with the mulligan (37%) for the overall 43%? I’m putting together a draft spreadsheet that I’ll send to you when finished.

        • TalesfromtheCards permalink

          Ooh, it’s been awhile, so hopefully I’m not misremembering, but I believe that I multiplied the probability (in fraction form) for the first event by the probability (in fraction form) of the second event.

        • Well, what he did, was combining the two probabilities for getting the specific card with the first six cards (32 %) and the seven “mulligan cards” (37 %). Then he stated that there’s only a 43 % chance of NOT pulling the desired card. I.e. a 57 % chance of pulling it over these two hands. I hope that made it clearer. ☺️

      • If you want to, I could probably send you such a spreadsheet. It wouldn’t be too much trouble, and it would be a fun assignment.

        • TalesfromtheCards permalink

          Yeah, I’d definitely be interested! I’d love to delve more into this probability stuff, but I haven’t had the time.

  18. Fantastic article! I just want to add some mathematical rules that will help simplify these formulas (I’m sorry if I use the wrong terms, my primary language isn’t English):
    1) You can put all these numbers in one fraction.
    2) Since all you do is multiply and divide, the order in which you do it is totally irrelevant.
    3) Remember that you can remove identical pairs of numerators/denominators.

    Applying these rules to the above-mentioned formula for the first seven different cards you see in a game will simplify the formula quite immensely:

    (43*42*41)/(50*49*48)=0.63 or 63 % as you quite correctly put it.

  19. For your opening hand (6 cards) and your mulligan hand (6 cards) plus your card (1 card) for the first resource phase, you can multiply the above-mentioned formula with the one for your first six cards:

    (43*42*41)/(50*49*48) * (44*43*42)/(50*49*48)

    Put everything into one big fraction, and you’ll get:

    (44*43*43*42*42*41)/(50*50*49*49*48*48)=0.425

    Meaning a (100-42.5=)57.5 % chance of actually drawing the card you’re looking for when including the mulligan option. Again, just like you stated.

    Actually most of the numbers in this formula could be divided into smaller factors, but that won’t make things easier to understand to most people. For those who are interested, the simplest, and therefore mathematically most “correct” fraction will be like this:

    (11*41*43^2)/(2^6*5^4*7^2) or written out in a fraction like most people are used to see it:

    11*41*43*43
    —————–
    6 4 2
    2 * 5 * 7

    I’m sorry if I made your head boil, but I warned you! 😄

  20. When it comes to “draw X cards” and “look at the top Y cards or your deck, put one [type] into hand/play”, it’s quite natural that the chances of hitting exactly what you want (IF you’re only looking for one particular card) is better in the latter example if Y is bigger than X, as you’ll see more cards (and vice versa). After all, you can pick from among all those revealed, and you only want one particular card. If you want to increase your options or try to get different combos going, you want as many cards in your hand as possible, of course.

    • TalesfromtheCards permalink

      Yeah, definitely true. This article is all about pulling a specific card for a deck and is really applicable to card-dependent decks. General card draw of course can be more useful for getting your deck going in a more general sense. It would be interesting to delve further into probabilities of drawing certain sets or types of cards and how card draw affects that.

  21. TwiceBorn permalink

    “Never tell me the odds!”

    — Han Solo, overheard at a LotR LCG table 🙂

    • TalesfromtheCards permalink

      Hahaha. The funny part of all this is that while I love probability and all this probability talk, I think I tend to be one of the more reckless, gambler-type players around. I love the thrill of the roll of the dice and the undefended attack!

  22. scott w. permalink

    This is a well-written and useful article. Thank you!

    • TalesfromtheCards permalink

      Thanks for reading! I’m glad you found it useful!

  23. Irbi permalink

    Really nice article. I love everything you do here on this blog.

    For the less math inclined readers, I would suggest explaining just a bit more how you got to the 57% chance. I got a little lost at that jump. For some reason I was trying to multiply 32% by 37% and got all sorts of confused 🙂

    You took the probability (63%) of not getting the card in your 6 carded hand and multiply that by the probability (68%) of not getting the card in your 7 carded hand.

    .63 * .68 = .43

    So you have a 43% chance of not drawing the card in either hand. Which means you have a 57% chance of getting what you want.

    • TalesfromtheCards permalink

      Thanks for the kind words! And thanks also for the math clarification for those readers who got similarly lost. It’s definitely not the most intuitive thing in the world and I had to brush up on my own probability knowledge when writing this article.

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