Probability of a Card Draw

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!

Glad it was helpful.

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

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
which adds up to 1.

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

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.

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!