Recommend
3 
 Thumb up
 Hide
14 Posts

BoardGameGeek» Forums » Gaming Related » General Gaming

Subject: Need Help With Card Drawing Probabilities rss

Your Tags: Add tags
Popular Tags: [View All]
Ben
United States
Ann Arbor
Michigan
flag msg tools
Avatar
mbmbmbmbmb
Calling all mathematicians! (Or other generally smart folk.)

I'm trying to think through a game, potentially for a review, and it would be very helpful for me to understand card draw probabilities.

The Setup:
108 card deck
16 each of 1s 2s and 3s
12 each of 4s 5s and 6s
8 each of 7s 8s and 9s

What I Need To Know:
In an average five card hand, how many pairs are likely?
What about a six-card hand?


I'm hoping this isn't too hard to people who know what they're doing.
geekgold for any help.
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
いい竹やぶだ!

South Euclid
Ohio
msg tools
Avatar
mbmbmb
Howdy! An interesting challenge, but I need to ask you a couple questions to get a mathematically precise description of the problem.

How do you wish to count 3+ of a kind? Three of a kind is often considered three pairs and four of a kind six pairs (as in cribbage). Those are the number of possible pairs you can draw from three and four cards, respectively. By extension, five of a kind would be 15 pairs. If you wish to count 3, 4, 5 and 6 of a kind all separately, the problem gets complicated.

Also, when you say "how many pairs are likely" do you mean what is the expected number (i.e., mean) number of pairs in a hand? Would you like the entire probability distribution, i.e., a list of the chances of 0 pairs, 1 pair, etc.?
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
いい竹やぶだ!

South Euclid
Ohio
msg tools
Avatar
mbmbmb
One more thing: Do you want an analytical (exact) answer, or is an approximation close enough? For example, it would take me quite a while to calculate all the probabilities mathematically, but I could write and run a simulation to estimate the chances to, say, three significant digits pretty easily.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Ben
United States
Ann Arbor
Michigan
flag msg tools
Avatar
mbmbmbmbmb
robigo wrote:
Howdy! An interesting challenge, but we first need a mathematically precise description of the problem.

How do you wish to count 3+ of a kind? Three of a kind is traditionally considered three pairs and four of a kind six pairs (as in cribbage). Those are the number of possible pairs you can draw from three and four cards, respectively. By extension, five of a kind would be 15 pairs.

Also, when you say "how many pairs are likely" do you mean what is the expected number (i.e., mean) number of pairs in a hand? Would you like the entire probability distribution, i.e., a list of the chances of 0 pairs, 1 pair, etc.?

Wow, so this is potentially more complicated than I thought.

First off, approximations are fine. My goal is to get a sense of the average hand (e.g. "nearly 60% of the time, you will start with one pair and three other distinct cards"). To that end, I would like to treat 3 of a kind as its own thing as with 4 of a kind (let me know if that just complicates things).

So, with a five card hand, I guess I'm asking for approximate odds of the following hands:

5 distinct cards.
One pair and three other distinct cards.
Two distinct pairs and one other card.
Three of a kind and two other distinct cards.
Four of a kind and one other card.

Does that make sense?
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
K Septyn
United States
Unspecified
Michigan
flag msg tools
SEKRIT MESSAGE SSSHHHHHHHHH
badge
Avatar
mbmbmbmbmb
robigo wrote:
Three of a kind is traditionally considered three pairs and four of a kind six pairs (as in cribbage). Those are the number of possible pairs you can draw from three and four cards, respectively. By extension, five of a kind would be 15 pairs.


I agree with everything except "traditionally considered". But I do suspect the OP is looking for "one pair" hands as are usually found in poker.
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
いい竹やぶだ!

South Euclid
Ohio
msg tools
Avatar
mbmbmb
Septyn wrote:
robigo wrote:
Three of a kind is traditionally considered three pairs and four of a kind six pairs (as in cribbage). Those are the number of possible pairs you can draw from three and four cards, respectively. By extension, five of a kind would be 15 pairs.


I agree with everything except "traditionally considered". But I do suspect the OP is looking for "one pair" hands as are usually found in poker.


Yeah—changed that to "often considered".
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
いい竹やぶだ!

South Euclid
Ohio
msg tools
Avatar
mbmbmb
chally wrote:

Wow, so this is potentially more complicated than I thought.

First off, approximations are fine. My goal is to get a sense of the average hand (e.g. "nearly 60% of the time, you will start with one pair and three other distinct cards"). To that end, I would like to treat 3 of a kind as its own thing as with 4 of a kind (let me know if that just complicates things).

So, with a five card hand, I guess I'm asking for approximate odds of the following hands:

5 distinct cards.
One pair and three other distinct cards.
Two distinct pairs and one other card.
Three of a kind and two other distinct cards.
Four of a kind and one other card.

Does that make sense?


Yep—five of a kind will be possible, too. That's not as many possibilities as I'd thought at first. The six-card case would include (where each number means "n of a kind of a unique rank"):

1-1-1-1-1-1
2-1-1-1-1
2-2-1-1
2-2-2
3-1-1-1
3-2-1
3-3
4-1-1
4-2
5-1
6
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
いい竹やぶだ!

South Euclid
Ohio
msg tools
Avatar
mbmbmb
Here's the results from a quick pilot study of 100,000 draws of 5 cards each. The numbers of each possible hand were:

Five different: 25466
One pair: 51277
Two pair: 12813
Three of a kind: 8608
Full house: 1201
Four of a kind: 618
Five of a kind: 17


(where "full house" is 3 + 2)

The estimated percentage of each kind of hand is, of course, just (the number drawn) / 100,000. The standard error of each estimate is (the square root of the number drawn) / 100,000. For example, the probability for three of a kind is 0.08608 ± 0.00093 (estimate ± SE). That's an error of a couple percent of the estimate, which is fairly decent.

The probability of 5 of a kind isn't very well approximated here, since there were only N instances. However, the exact probability is easy to calculate for 5 of a kind. The number of ways to draw five 1s, 2s or 3s is 16!/(5!11!) = 4368 each, of 4s, 5s or 6s = 12!/(5!7!) = 792, and of 7s, 8s or 9s = 8!/(5!3!) = 56 each. And the total number of ways to draw any hand (including identical-looking ones) is 108!/(5!103!) = 111,469,176. So, the probability of drawing 5 of a kind is

(3 * 4368 + 3 * 792 + 3 * 56) / 111469176 = 15648/111469176 ~ 0.00014

That's within error of the estimate above, 0.00017 ± 0.00004.
2 
 Thumb up
11.01
 tip
 Hide
  • [+] Dice rolls
いい竹やぶだ!

South Euclid
Ohio
msg tools
Avatar
mbmbmb
jmzero, I bow before your superior probability calculatin' skillz. How did you do that?

Edit: Never mind—I think I've figured it out.

Edit 2: Ah, that's sneaky. Somewhat different, and more elegant, than my solution.
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
K Septyn
United States
Unspecified
Michigan
flag msg tools
SEKRIT MESSAGE SSSHHHHHHHHH
badge
Avatar
mbmbmbmbmb
jmzero wrote:
For 5 cards, the odds are:

2-1-1-1 - 0.512380427033938


Excellent work, sir. Your number for the one-pair case matches what I found. My slow method was to categorize 1-3, 4-6, 7-9 into different classes, then count the number of ways to make a one-pair hand for each of 27 different class combinations. I stopped back in to see if anyone beat me to the punch, and thank goodness someone did.
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Ben
United States
Ann Arbor
Michigan
flag msg tools
Avatar
mbmbmbmbmb
Thanks, all! This is tremendously helpful.
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
いい竹やぶだ!

South Euclid
Ohio
msg tools
Avatar
mbmbmb
Thank you, and good luck!
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Luis Fernandez
Venezuela
Caracas
Miranda
flag msg tools
badge
Avatar
mbmbmbmbmb
two word for you

Card Weaving!

Nah that´s trap, but if you place the cards in convenient places in the deck and then shuffle well, they would tend to have very variable positions.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Front Page | Welcome | Contact | Privacy Policy | Terms of Service | Advertise | Support BGG | Feeds RSS
Geekdo, BoardGameGeek, the Geekdo logo, and the BoardGameGeek logo are trademarks of BoardGameGeek, LLC.