# Probability of a Card Draw

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jbmoyer
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Joined: 04/27/2012

Hello, I am developing a Card Game and was wondering how to determine the probability of drawing certain cards. I am sure there is probably simple math for it. For example, during the course of a game how often will a Monster, Loot or Trap be drawn? I know I can weight the odds by putting in more Monsters for example, but how can you determine a good mix?

Thanks, this forum is awesome.

Brett

MarkKreitler
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Joined: 11/12/2008
Hope this is what you're after

Hey Brett,

It sounds like you're after some simple statistics formulae. I'll throw some your way. If these are too simple or otherwise not useful, I apologize. If you can be more specific about your needs, we can give you better answers.

The easiest way to compute the probability is to take the total number of cards of the type you want to draw (like Monsters) and divide by the total number of cards remaining in the deck. So, if you have 20 monsters cards and 30 non-monster cards in your deck,

The odds of drawing a monster is: 20 / 50, or 2/5.
The odds of drawing something besides a monsters is: 30/50, or 3/5.

The tricky part comes as you play for awhile and the deck has been picked through. The same math applies, but you'll get different results depending on what players have drawn up to that point.

For example, suppose in game #1, players have drawn 8 monsters and 5 "something elses." Now, the chance to draw a monster is:

12 monsters left / 37 total cards = 12 / 37 which is approximately 12 / 36 = 1 / 3

In game #2, players have drawn 3 monsters and 10 "something elses", so now the probability of a monster draw is:

17 monsters left / 37 total cards = which is approximately 18 / 36 = 1 / 2.

In general, it doesn't help the design much to consider the middle-of-the-game situations because so much depends on what happened up to that point, and that's up to chance.

How can you determine a good mix for your monsters, traps, and loot? Start with how you want the game to play. Should players get monsters about 1/2 the time, with the other half evenly split between loot and traps? Then:

Monster chance = 1/2 the time
Trap chance = 1/2 the time that isn't monsters * 1/2 of the "isn't monsters" time = 1/2 * 1/2 = 1/4
Loot chance = 1/2 the time that isn't monsters * 1/2 of the "isn't monsters" time = 1/2 * 1/2 = 1/4

Here's an easy check to make sure you've computed your probabilities the correct way: if you add up the chances of every possible outcome in your game, they should equal '1'.

So: 1/2 + 1/4 + 1/4 = 1/2 + 1/2 = 1, and the above probabilities add up.

To translate this back to the number of cards, just multiply the probability of each outcome times the number of cards in your deck. So, if you're using a 52 card deck:

# of Monster cards = 52 total cards * 1/2 chance of drawing a monster = 52 * 1/2 = 26.
# of Loot cards = 52 total cards * 1/4 chance of drawing loot = 52 * 1/4 = 13.
# of Trap cards = 52 total cards * 1/4 chance of drawing a trap = 52 * 1/4 = 13.

Another safety check: adding up all the individual types of cards should equal the total number of cards:

26 Monster cards + 13 Loot cards + 13 Trap cards = 52 -- you're good to go.

Once you have this math down, you can easily tune your game to your whims. Want a tough game with 4 monster cards and 3 traps for every piece of loot?

Odds of drawing a monster = 4 / (4 + 3 + 1) = 4 / 8 = 1/ 2
Odds of drawing a trap = 3 / (4 + 3 + 1) = 3 / 8
Odds of drawing loot = 1 / (4 + 3 + 1) = 1/8

Check: odds of something happening (monster or trap or loot) = 1/2 + 3/8 + 1/8 = 1/2 + 4/8 = 1/2 + 1/2 = 1 CHECK

Want a 64 card deck?

Monster cards = 64 * 1/2 = 32 cards
Trap cards = 64 * 3/8 = 24 cards
Loot cards = 64 * 1/8 = 8 cards

Check: total cards = 32 + 24 + 8 = 32 + 32 = 64 CHECK

That should be enough to get you started.

Happy balancing!

jbmoyer
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Joined: 04/27/2012
Thank you Mark

Wow, thank you. Far more then I expected. You have given me plenty to chew on. To give you more detail, but I think your explanation was excellent. My game is an RPG where you place room tiles (cards) and then a certain number of Encounter Cards(Monsters, Loot, Traps) on that room. I want to make sure I have a good balance of Encounter cards so that the game is even but challenging and overall fun of course. I didn't want the player to draw a monster each turn, but also didn't want them always be drawing loot and stock piling weapons etc...

Thanks so much Mark greatly appreciated.

Brett

MarkKreitler
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Joined: 11/12/2008
Great!

I love that kind of game, by the way. Good luck with it -- I hope you'll post more as your design comes along.

Mark

jbmoyer
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Joined: 04/27/2012
Clarity on Math

Call me Dumb, Please! :) But I dont think I quite understand what you stated here:

Monster chance = 1/2 the time (I get this)
Trap chance = 1/2 the time that isn't monsters * 1/2 of the "isn't monsters" time = 1/2 * 1/2 = 1/4
Loot chance = 1/2 the time that isn't monsters * 1/2 of the "isn't monsters" time = 1/2 * 1/2 = 1/4

How does 1/2 * 1/2 = 1/4? Should that equal 1?
- RETRACTION, I get it now, that I retyped it, Check Me!
So 1/2 * 1/2 = 1/4 of the time LEFT and so Loot and Traps each are a 1/4 thus in (TOTAL 1) with 1/2 for Monsters!. Oh Boy. :) I was just seeing 1/2 * 1/2 surly dosent equal 1/4.

and here:

So: 1/2 + 1/4 + 1/4 = 1/2 + 1/2 = 1, and the above probabilities add up.

I am just plain confused, too many = signs I think.

I spent the night tonight coming up with break downs of Monsters, Loot, NPC's, Rooms and total cards based on some of the math you gave me, I think I am on to something, but of course I need to play test it to death. Mock/Generic cards tomorrow. Thanks

Brett

MarkKreitler
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Joined: 11/12/2008
I can overcomplicate anything!

Hey Brett,

No worries. I'm best known around here for my inability to communicate. You should try reading my rule books. :(

> So 1/2 * 1/2 = 1/4 of the time LEFT and so Loot and Traps each are a 1/4 thus in (TOTAL 1) with 1/2 for Monsters!. Oh Boy. :) I was just seeing 1/2 * 1/2 surly dosent equal 1/4.

Exactly right.

> So: 1/2 + 1/4 + 1/4 = 1/2 + 1/2 = 1, and the above probabilities add up.
>I am just plain confused, too many = signs I think.

Yeah, that is a lot of = signs. Let me rewrite it in English.

So a half plus two quarters adds up to 1.

Or: 1/2 + 1/4 + 1/4
is the same as 1/2 + 1/2

Hope that's more clear!

Making mock cards is the best part. I'm envious. I get to work this weekend. :( Next weekend, though, I'll be mocking up cards of my own. :D

MarkKreitler
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Joined: 11/12/2008
Interesting alternative

Hey Brett,

I can tell from your comments that you've grasped the math (and quickyl!), but I thought of a cool alternative method that's "mathless".

1) Get poker chips of 3 different colors.
2) Let 1 color represent monsters, another traps, the third color is loot.
3) Now build the smallest possible stack consisting of the different colors in the proportion you think looks "about right."

For example: white is monsters, red is traps, blue is loot. Build a stack with a lot of whites, a fair amount of reds, and a few blues (for a tough game). When you think you have the smallest possible good stack, test it out: dump the chips into a bag and draw until you've picked them all out. If it *does* feel good, repeat this test a couple times to be sure.

Once you've got a mix you like, just pretend each chip represents a fixed number of cards (like 5). Then you can figure out how big your deck is. For example, suppose you found the mix you like is 7 whites (monsters), 2 reds (traps) and 3 blues (loot). Now, let each chip represent 5 cards, so your final deck will have

7 x 5 = 35 monsters
2 x 5 = 10 traps
3 x 5 = 15 loot cards

for a 60 card deck.

If that's too many, let each chip represent 4 cards, or 3, or whatever, until you have the size you want. Same thing in reverse if you want a bigger deck.

OK, so I lied and it's not quite mathless, but it's easier than crunching fractions and gives you an immediate feel for the game play. I think I'm going to try this myself on my next card game...

KAndrw
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Joined: 08/20/2008
The hypergeomdist function in

The hypergeomdist function in excel is very useful for such things:

=Hypergeomdist( num_copies_wanted, num_cards_drawn, num_copies_in_deck, num_cards_in_deck )

will tell you the probability of drawing exactly [num_copies_wanted] copies of your target card.

jbmoyer
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Joined: 04/27/2012
I Love It!

Wow, thanks again, I love the idea of drawing poker chips and Thanks KAndrw, I figured there must be some sort of formula out there. World is built on math isn't it :)

KAndrw
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Joined: 08/20/2008
MarkKreitler wrote:Once

MarkKreitler wrote:
Once you've got a mix you like, just pretend each chip represents a fixed number of cards (like 5). Then you can figure out how big your deck is. For example, suppose you found the mix you like is 7 whites (monsters), 2 reds (traps) and 3 blues (loot). Now, let each chip represent 5 cards, so your final deck will have

7 x 5 = 35 monsters
2 x 5 = 10 traps
3 x 5 = 15 loot cards

While a fun way of doing things and a good rough guide, be aware that this quickly becomes inaccurate.

Consider the traps - with 12 chips including 2 traps, you have a 1/6 chance of drawing a trap as your first chip, then a 1/11 chance of drawing a second trap. That means a 1/66 chance of getting two traps as your first two draws.

Increase the pool by five times, and the chance of the initial trap remains the same - 1/6. However, the chance of a second trap increases to 9/59, giving you approximately 1/39 of drawing two traps in a row.

I'm not in any way saying that it's a bad idea, but make sure that you don't rely on the results from the small scale test when making probablistic claims about the final product!

MarkKreitler
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Joined: 11/12/2008
Good point

That's a good point.

It could be I interpreted Brett's initial question incorrectly, but I assumed he wanted a method that let him generate a deck with a certain "feel." By feel, I mean "By the time I fight 'm' monsters, I should get about 'l' loot and hit 't' traps." That question implies averaging over many samples to get a general behavior.

In that case, the poker chip method holds up. Working with the smaller sample excludes possibilities extant in the larger sample (like drawing all 10 traps in a row), but has the same broad behavior.

If you want to answer questions about particular events -- like the probability of drawing a Monster from a deck of M + T + L cards from which m monsters, t traps, and l loot has been drawn -- you can't generalize from the smaller sample to the large. On the other hand, analyzing that single event isn't useful for determining the broad behavior of the game system. To do that, you'd have to sum over all possible events, which would bring you back to the general behavior that's captured equally well by the smaller sample.

I think.

Statistics was never my strong point. :P

jbmoyer
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Joined: 04/27/2012
You Got It Right, Sir!

Thank you very much Mark you understood what I was after and gave me more then plenty to think on. I did a quick play test today and I have too many traps, they came up far to often to be safe :) so I think I will tweak out the traps and bring in more loot. :)

Thanks
Brett

MarkKreitler
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Joined: 11/12/2008
Woot!

jbmoyer wrote:
Thank you very much Mark you understood what I was after and gave me more then plenty to think on. I did a quick play test today and I have too many traps, they came up far to often to be safe :) so I think I will tweak out the traps and bring in more loot. :)

Thanks
Brett

Awesome, Brett! Glad I could help.

Sounds like the play-testing is doing exactly what it should. Hope you'll post some news when the game hits its groove!