Please check my math
March 2nd, 2017 at 5:19:50 PM
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I am having a brain fart or something, could someone please check my math?
We are drawing 8 cards from a 52 card deck and we are unconcerned about straights and flushes -we are just interested in the frequency of pairing, namely these two specific cases:
1. 3 pairs + 2 singletons (i.e., 5 distinct ranks)
2. 4 pairs
For 3 pairs + 2singletons, I calculate P= 5.9105%.
= [ c(13,3) * c(4,2)^3 *c(10,2) * c(4,1)^2 ] / c(52,8)
For 4 pairs, I calculate P =0.1231%
= [ c(13,4) * c(4,2)^4 ] / c(52,8)
I feel like a senile dummy for asking, but is that correct? The reason I ask is that, given these numbers, 3 pairs (+ 2 singletons) is more frequent than 4 pair by 48:1, and a friend has asked me why is it 48:1, shouldn't it be 12:1? And I can't quite summon an answer to that.
We are drawing 8 cards from a 52 card deck and we are unconcerned about straights and flushes -we are just interested in the frequency of pairing, namely these two specific cases:
1. 3 pairs + 2 singletons (i.e., 5 distinct ranks)
2. 4 pairs
For 3 pairs + 2singletons, I calculate P= 5.9105%.
= [ c(13,3) * c(4,2)^3 *c(10,2) * c(4,1)^2 ] / c(52,8)
For 4 pairs, I calculate P =0.1231%
= [ c(13,4) * c(4,2)^4 ] / c(52,8)
I feel like a senile dummy for asking, but is that correct? The reason I ask is that, given these numbers, 3 pairs (+ 2 singletons) is more frequent than 4 pair by 48:1, and a friend has asked me why is it 48:1, shouldn't it be 12:1? And I can't quite summon an answer to that.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
March 2nd, 2017 at 7:04:03 PM
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Your math is correct - 3 pairs and 2 singletons are 48 times as likely as 4 pairs.
As to why it's 48x and not 12x, I think it has something to do with the fact that, when you "add the fourth pair" to three existing pairs, you may be counting the four pairs four times (e.g. if you have pairs of Jacks, Queens, and Kings, and you add a pair of Aces, JQK+A is the same as JQA+K, JKA+Q, and QKA+J).
As to why it's 48x and not 12x, I think it has something to do with the fact that, when you "add the fourth pair" to three existing pairs, you may be counting the four pairs four times (e.g. if you have pairs of Jacks, Queens, and Kings, and you add a pair of Aces, JQK+A is the same as JQA+K, JKA+Q, and QKA+J).
March 2nd, 2017 at 7:22:03 PM
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How do you count the case of quads drawn? ie: 5555992? Is this situation 3 pairs or not?
- your comment indicates "5 distinct ranks" which would mean no? have you accounted for this situation in your math?
- your comment indicates "5 distinct ranks" which would mean no? have you accounted for this situation in your math?
March 2nd, 2017 at 8:53:13 PM
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Quote: rsactuaryHow do you count the case of quads drawn? ie: 5555992? Is this situation 3 pairs or not?
- your comment indicates "5 distinct ranks" which would mean no? have you accounted for this situation in your math?
Of course, quads (and trips) are distinctive from pairs and were not included in my calculation.
I calculate that you will draw Quads+Pair+2 singletons with a frequency of 0.1095% -so that is slightly less frequent than 4 pair
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
March 2nd, 2017 at 8:54:34 PM
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Quote: ThatDonGuyYour math is correct - 3 pairs and 2 singletons are 48 times as likely as 4 pairs.
As to why it's 48x and not 12x, I think it has something to do with the fact that, when you "add the fourth pair" to three existing pairs, you may be counting the four pairs four times (e.g. if you have pairs of Jacks, Queens, and Kings, and you add a pair of Aces, JQK+A is the same as JQA+K, JKA+Q, and QKA+J).
What is really frightening is that I understood that answer. Thank you.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
