From the previous discussion, we assumed that the players play any kind of starting hands. Or we should say that the probability distribution is the same for each starting hand.
If a player is selective on the starting hands with considering each player's chips, strategy, style, and position, the probability of each starting hand will be different.
So the estimated probability of the possible 5 card combination (after the flop) is based on the fact (the paired flop) and the probability distribution of the player's starting hand.
For example, if the flop is 99K, and from the past experience player X only plays starting hand 99. Then we would know the probability of four of a kind is 100% for player X.
9/22/2014
Texas Hold'em Poker Probability - Paired Flop cont.
With your two hole cards, the other players might have...
Possible starting hands: 1081=47*46/2=C(47,2)
Assuming the flop is 99K,
a) if you flop a three of a kind with 9x
Four of Kind: 0
Three of a Kind or Full House: 1*46+3=49 type 1: 9y, type 2: KK
Two Pair: 3*43+10*6+3=192 type 1: Ky, type 2: yy, type 3: xx
Others: 43*42/2-(10*6+3)=840 any combination without 9 and K without one x card(43*42/2), minus yy and xx pairs.
There is about 22.3% (1 in 4.5) having a two pair or better.
b) if you have a paired starting hand with QQ
Four of Kind: 1
Three of a Kind or Full House: 2*45+3=93 type 1: 9y, type 2: KK
Two Pair: 3*42+10*6+1=187 type 1: Ky (y is any card <> K9), type 2: yy (y<>Q), type 3: QQ
Others: 42*41/2-(10*6+1)=800 any combination without 9s and Ks and two Q cards(42*41/2), minus pairs.
There is about 26% (1 in 4) having a two pair or better. It's slight better than the original 25% that doesn't consider your hole cards.
c) if you flop a two pair with Kx
Four of Kind: 1
Three of a Kind or Full House: 2*45+1=91 type 1: 9y, type 2: KK
Two Pair: 2*43+10*6+3=149 type 1: Ky, type 2: yy ( y not in (9,K,x)), type 3: xx
Others: 43*42/2-(10*6+3)=840
There is about 22.3% (1 in 4.5) having a two pair or better.
d) if your starting hand is XY, X<>Y and not in (9,K)
Four of Kind: 1
Three of a Kind or Full House: 2*45+3=93 type 1: 9y, type 2: KK
Two Pair: 3*42+9*6+2*3=186 type 1: Kz (z is any card not in (9,K)), type 2: zz (z not in (X,Y)), type 3: xx ( x is X or Y)
Others: 42*41/2-(9*6+2*3)=801
There is about 26% (1 in 4.5) having a two pair or better.
Possible starting hands: 1081=47*46/2=C(47,2)
Assuming the flop is 99K,
a) if you flop a three of a kind with 9x
Four of Kind: 0
Three of a Kind or Full House: 1*46+3=49 type 1: 9y, type 2: KK
Two Pair: 3*43+10*6+3=192 type 1: Ky, type 2: yy, type 3: xx
Others: 43*42/2-(10*6+3)=840 any combination without 9 and K without one x card(43*42/2), minus yy and xx pairs.
There is about 22.3% (1 in 4.5) having a two pair or better.
b) if you have a paired starting hand with QQ
Four of Kind: 1
Three of a Kind or Full House: 2*45+3=93 type 1: 9y, type 2: KK
Two Pair: 3*42+10*6+1=187 type 1: Ky (y is any card <> K9), type 2: yy (y<>Q), type 3: QQ
Others: 42*41/2-(10*6+1)=800 any combination without 9s and Ks and two Q cards(42*41/2), minus pairs.
There is about 26% (1 in 4) having a two pair or better. It's slight better than the original 25% that doesn't consider your hole cards.
c) if you flop a two pair with Kx
Four of Kind: 1
Three of a Kind or Full House: 2*45+1=91 type 1: 9y, type 2: KK
Two Pair: 2*43+10*6+3=149 type 1: Ky, type 2: yy ( y not in (9,K,x)), type 3: xx
Others: 43*42/2-(10*6+3)=840
There is about 22.3% (1 in 4.5) having a two pair or better.
d) if your starting hand is XY, X<>Y and not in (9,K)
Four of Kind: 1
Three of a Kind or Full House: 2*45+3=93 type 1: 9y, type 2: KK
Two Pair: 3*42+9*6+2*3=186 type 1: Kz (z is any card not in (9,K)), type 2: zz (z not in (X,Y)), type 3: xx ( x is X or Y)
Others: 42*41/2-(9*6+2*3)=801
There is about 26% (1 in 4.5) having a two pair or better.
Texas Hold'em Poker Probability - Paired Flop
As we might see a paired flop about 17% of time, let's take a look at what kind of hands players might have.
Possible starting hands: 1176=49*48/2=C(49,2) we can see here the possible starting hands are updated based on the flop (based on the fact).
Four of a Kind: 1
Three of a Kind or Full House: 2*47 + 3 = 97 type 1: pick one card with the same number as the paired card(2) and pick anyone from the remaining card without the same number(47), type 2: pick two cards from with the same number as the unpaired card(3)
Two Pair: 198=3*44+11*6 type 1: pick one with the same number as the unpaired card(3) and pick anyone from the remaining card without the number in the flop(44), type 2: pick any paired card without the number in the flop (11*6)
Others: 880=44*40/2=C(11,2)*4^2 pick any unpaired card without the number in the flop
There is about 25% (1 in 4) having a two pair or better.
* * *
The previous 3 cards or 5 cards analysis give players an idea of what might happen during the game before you see the flop or give a general idea of the distribution of the combinations.
But after you see the flop, the situation is changed. You will have to update the distribution of the combinations based on the flop or based on the fact.
By the way, you also have two cards at hand, so what the other players might have will be different from the previous calculated numbers.
Possible starting hands: 1176=49*48/2=C(49,2) we can see here the possible starting hands are updated based on the flop (based on the fact).
Four of a Kind: 1
Three of a Kind or Full House: 2*47 + 3 = 97 type 1: pick one card with the same number as the paired card(2) and pick anyone from the remaining card without the same number(47), type 2: pick two cards from with the same number as the unpaired card(3)
Two Pair: 198=3*44+11*6 type 1: pick one with the same number as the unpaired card(3) and pick anyone from the remaining card without the number in the flop(44), type 2: pick any paired card without the number in the flop (11*6)
Others: 880=44*40/2=C(11,2)*4^2 pick any unpaired card without the number in the flop
There is about 25% (1 in 4) having a two pair or better.
* * *
The previous 3 cards or 5 cards analysis give players an idea of what might happen during the game before you see the flop or give a general idea of the distribution of the combinations.
But after you see the flop, the situation is changed. You will have to update the distribution of the combinations based on the flop or based on the fact.
By the way, you also have two cards at hand, so what the other players might have will be different from the previous calculated numbers.
9/20/2014
Texas Hold'em Poker Probability - Flop (3 cards)
So what exactly we can see on the flop (3 cards).
There are 22100 combinations, 22100=C(52,3).
Straight Flush: 48=12*4 from A23 to QKA with 4 suits.
Three of a Kind: 52 = 13*4 pick one number from A to K with 4 different suit combinations.
Straight: 720=12*(4^3-4) from A23 to QKA with 4^3 suit combinations minus flush(4).
Flush: 1096=C(13,3)*4-48 pick 3 different numbers from A to K with 4 suits minus straight flush(48).
Pair: 3744=13*6*48 pick one number from A to K with 6 suit combination, pick one from the remaining 48 cards.
Others: 16440= (C(13,3)-12)*(4^3-4) pick 3 different numbers from A to K minus straight, without flush (4^3-4).
So it's about 17% ( 1 in 6 times) to see a pair on the flop.
It's about 26% ( 1in 4 times) to see a pair or better or flush/straight on the flop.
***
We may also want to know what about 2 card flush on the flop, or some kind of straight like patterns on the flop, such as 679, 689, or 579.
2 Card Flush: 12168=C(13,2)*13*3*4 pick two different numbers from A to K for the 2 card flush (C(13,2)), pick another one from A to K for the different suit (13), the suit combination is 4*3.
12168 includes one pair and straight.
So to see a 2 card flush or flush is about 60%, it's quite high, almost more than 1 in 2 games.
What about 2 Card Flush without pair or straight?
It's 9864=(C(13,3)-12)*P(4,2)*3 pick three different numbers from A to K and minus straight(12), the suit combination is P(4,2)*3 (pick two different suits, one for two card flush, one for the other)
***
For straight like combinations, let's consider 3 cards with different suits first
A24~JQA: 264=11*P(4,3) 3 cards all have different suits, P(4,3)
A34~JKA: 264=11*P(4,3)
A24~JQA: 660=11*(P(4,3)+P(4,2)*3) = 11*(4^3-4)
A34~JKA: 660=11*(4^3-4)
There are 22100 combinations, 22100=C(52,3).
Straight Flush: 48=12*4 from A23 to QKA with 4 suits.
Three of a Kind: 52 = 13*4 pick one number from A to K with 4 different suit combinations.
Straight: 720=12*(4^3-4) from A23 to QKA with 4^3 suit combinations minus flush(4).
Flush: 1096=C(13,3)*4-48 pick 3 different numbers from A to K with 4 suits minus straight flush(48).
Pair: 3744=13*6*48 pick one number from A to K with 6 suit combination, pick one from the remaining 48 cards.
Others: 16440= (C(13,3)-12)*(4^3-4) pick 3 different numbers from A to K minus straight, without flush (4^3-4).
So it's about 17% ( 1 in 6 times) to see a pair on the flop.
It's about 26% ( 1in 4 times) to see a pair or better or flush/straight on the flop.
***
We may also want to know what about 2 card flush on the flop, or some kind of straight like patterns on the flop, such as 679, 689, or 579.
2 Card Flush: 12168=C(13,2)*13*3*4 pick two different numbers from A to K for the 2 card flush (C(13,2)), pick another one from A to K for the different suit (13), the suit combination is 4*3.
12168 includes one pair and straight.
So to see a 2 card flush or flush is about 60%, it's quite high, almost more than 1 in 2 games.
What about 2 Card Flush without pair or straight?
It's 9864=(C(13,3)-12)*P(4,2)*3 pick three different numbers from A to K and minus straight(12), the suit combination is P(4,2)*3 (pick two different suits, one for two card flush, one for the other)
***
For straight like combinations, let's consider 3 cards with different suits first
A24~JQA: 264=11*P(4,3) 3 cards all have different suits, P(4,3)
A34~JKA: 264=11*P(4,3)
A35~TQA: 240=10*P(4,3)
If we include 2 card flush but not 3 card flush,
A24~JQA: 660=11*(P(4,3)+P(4,2)*3) = 11*(4^3-4)
A34~JKA: 660=11*(4^3-4)
A35~TQA: 600=10*(4^3-4)
So, we might see straight or straight-like flops about 12% ( 1 in 8 times).
So, we might see straight or straight-like flops about 12% ( 1 in 8 times).
Texas Hold'em Poker Probability - Flop (5 cards) cont.
Rather than the previous 5 card combinations, players might also be interested in knowing how often we get open-ended straight or 4 card flush after the flop just shown to us.
4 Card Flush: 111540=C(13,4)*4*(52-13) pick 4 different numbers from A to K with 4 suits, and pick one from the remaining card without the same suit(52-13).
Open Ended Straight: 60480=9*(13-6)*(4^5-4*3*5-4) from 2345 to TJQK and pick one from the remaining number without making straight (13-6), the total suit combination 4^5 minus 4-card-flush(4*3*5) minus flush(4).
***
After the flop is shown, there is about 1 in 13 players to have something better than one pair.
But if we also include the 4 card flush and open-ended straight, then it's about 1 in 7.
So if there might be 7 players into a game to see the flop, you would better be prepared to fight down through the river.
4 Card Flush: 111540=C(13,4)*4*(52-13) pick 4 different numbers from A to K with 4 suits, and pick one from the remaining card without the same suit(52-13).
Open Ended Straight: 60480=9*(13-6)*(4^5-4*3*5-4) from 2345 to TJQK and pick one from the remaining number without making straight (13-6), the total suit combination 4^5 minus 4-card-flush(4*3*5) minus flush(4).
***
After the flop is shown, there is about 1 in 13 players to have something better than one pair.
But if we also include the 4 card flush and open-ended straight, then it's about 1 in 7.
So if there might be 7 players into a game to see the flop, you would better be prepared to fight down through the river.
Texas Hold'em Poker Probability - Flop (5 cards)
There are 3 cards on the flop. Including your two cards, there are total 5 cards.
Let's look at the 5 card combination first.
The total combination of 5 cards is 2,598,960= 52*51*50*49*48/5!=C(52,5)
Royal Flush: 4
Straight Flush: 36=10*4-4 from A2345 to TJQKA minus Royal Flush
Four of a Kind: 624=13*12*4 from AAAA* to KKKK*, * could be any of the other 48 cards
Full House: 3744=P(13,2)*P(4,2) pick two different number from A to K for the 3 cards and 2 cards and with 4*3 suit combinations.
Flush: 5108=C(13,5)*4-36-4 pick five different numbers from A to K, four different suits, minus straight.
Straight: 10200=10*(4^5-4) from A2345 to TJQKA with any kind of suit combination minus flush.
Three of a Kind: 54912=13*4*C(48,2)-3744 from AAA to KKK with 4 different suit combination, and pick any two cards from the remaining 48 cards, then minus full house.
Two Pairs: 123552=C(13,2)*6*6*11*4 pick two different numbers from A to K, each number has 6 suit combination, and pick one from the remaining card without the same number(11*4).
One Pair: 1098240=13*6*C(12,3)*4^3 pick one number from A to K with 6 suit combination, then pick 3 numbers from the remaining 12 numbers (each number has 4 suit choices).
Others: 1302540=(C(13,5)-10)*(4^5-4) pick five different numbers from A to K minus the straight combination(10), there are 4^5 suit combination minus flush(4).
The probability to get one pair or better after the flop is about 50%.
How to VERIFY: Play online poker and keep the record of the hands with the flop. If you fold before the flop, you simply take that record out.
Let's look at the 5 card combination first.
The total combination of 5 cards is 2,598,960= 52*51*50*49*48/5!=C(52,5)
Royal Flush: 4
Straight Flush: 36=10*4-4 from A2345 to TJQKA minus Royal Flush
Four of a Kind: 624=13*12*4 from AAAA* to KKKK*, * could be any of the other 48 cards
Full House: 3744=P(13,2)*P(4,2) pick two different number from A to K for the 3 cards and 2 cards and with 4*3 suit combinations.
Flush: 5108=C(13,5)*4-36-4 pick five different numbers from A to K, four different suits, minus straight.
Straight: 10200=10*(4^5-4) from A2345 to TJQKA with any kind of suit combination minus flush.
Three of a Kind: 54912=13*4*C(48,2)-3744 from AAA to KKK with 4 different suit combination, and pick any two cards from the remaining 48 cards, then minus full house.
54912=13*4*C(12,2)*4^2 from AAA to KKK with 4 different suit combination, and pick any two different numbers from the remaining 12 numbers (each number has 4 suit choices).
Two Pairs: 123552=C(13,2)*6*6*11*4 pick two different numbers from A to K, each number has 6 suit combination, and pick one from the remaining card without the same number(11*4).
One Pair: 1098240=13*6*C(12,3)*4^3 pick one number from A to K with 6 suit combination, then pick 3 numbers from the remaining 12 numbers (each number has 4 suit choices).
Others: 1302540=(C(13,5)-10)*(4^5-4) pick five different numbers from A to K minus the straight combination(10), there are 4^5 suit combination minus flush(4).
The probability to get one pair or better after the flop is about 50%.
How to VERIFY: Play online poker and keep the record of the hands with the flop. If you fold before the flop, you simply take that record out.
Texas Hold'em Poker Probability - Starting Hand
Poker is an imperfect information game, and it's all about the theory of probability. The key to win is based on how we can use the probability to gain an edge with some call/raise skills.
In Texas Hold'em, you get two cards first, then the flop, the turn, and the river card. In such a process, we can find out the probability of the starting hand (2 cards), the flop(3/5 cards), the turn (4/6 cards), and the river (5/7 cards).
If we simplify it a little bit, we focus on the starting hand, the flop, and the river. We would like to know the 2-card, 3-card, 5-card, and 7-card probability of the combination of the cards.
There are 52 cards.
For the starting hand,
the number of the combination of the starting hand is 1,326 = 52*51/2.
pair - 78 = 52*3/2
suited - 312 = 52*12/2
others - 936 = 52*(51-3-12)/2
pair : suited : others = 1:4:12
You have about 30% probability to get a pair(6%) or suited(24%) starting hand.
How to VERIFY: You can go to play online poker and keep a record of your starting hands to verify the result.
6/11/2014
貶低新台幣匯率的後遺症
台灣央行長期為保護外銷產業而刻意貶低新台幣匯率 已使得台灣從四小龍變成蟲 也許有人會認為如果不這樣做 今天台灣會更糟
但事實上 當你刻意照顧某產業時 往往就會犧牲其他人的利益 長期貶低新台幣會有以下幾種後遺症
仰賴進口物品的消費者成本增加 例如購車的人 台灣並沒有製造汽車主要零組件如引擎的能力 也缺乏設計的團隊 幾乎全是進口車 台灣人購車的成本以所必須付出的工作時間而言相對增加
出國觀光留學研習出訪的成本較高 以相同的勞力付出 卻只能做較少的消費 而觀光留學等對提升台灣競爭力長期而言有相當大的幫助
國內內需產業服務人員的國際薪資水準偏低 台灣一個大學生的起薪(NTD165/hr)不如美國加州的最低薪資(NTD240/hr) 這再怎麼樣也說不過去 有人會說台灣物價較便宜 問題是美國的物價在已開發國家中也是算低的 以洛杉磯的物價水準 台灣大學生起薪約為(NTD193/hr) 這還是遠低於加州的最低薪資
產業轉型不易 在保護政策下 產業較不願意做技術上或管理上的投資開發 只願意在削減人力成本上動腦筋
以總體經濟學的理論 要刻意貶低台幣 央行就必須付出相對的代價 這也是台灣大眾要負擔的成本 而<長期>貶低新台幣的成本累積 早晚有一天會壓垮央行 抑或是壓垮台灣的競爭力
貶低新台幣就好像貶低台灣的產品價格與勞動成本 最主要的就是犧牲台灣人民因勞動所得到的報酬
當然有人會認為筆者所言並不一定會發生 或舉出貶低新台幣的好處
沒錯 這些後遺症並不一定會發生 但事實上在台灣就有這些問題發生 有時候很難從匯率去比較 但如果從購買力指數 或大學生與專業人員的國際薪資水平 就可看出來台灣有很嚴重的問題所在 台灣的人才外流與外留已不是新鮮事 人才的流動可不是政府能夠控制的
醒醒吧 央行!
但事實上 當你刻意照顧某產業時 往往就會犧牲其他人的利益 長期貶低新台幣會有以下幾種後遺症
仰賴進口物品的消費者成本增加 例如購車的人 台灣並沒有製造汽車主要零組件如引擎的能力 也缺乏設計的團隊 幾乎全是進口車 台灣人購車的成本以所必須付出的工作時間而言相對增加
出國觀光留學研習出訪的成本較高 以相同的勞力付出 卻只能做較少的消費 而觀光留學等對提升台灣競爭力長期而言有相當大的幫助
國內內需產業服務人員的國際薪資水準偏低 台灣一個大學生的起薪(NTD165/hr)不如美國加州的最低薪資(NTD240/hr) 這再怎麼樣也說不過去 有人會說台灣物價較便宜 問題是美國的物價在已開發國家中也是算低的 以洛杉磯的物價水準 台灣大學生起薪約為(NTD193/hr) 這還是遠低於加州的最低薪資
產業轉型不易 在保護政策下 產業較不願意做技術上或管理上的投資開發 只願意在削減人力成本上動腦筋
以總體經濟學的理論 要刻意貶低台幣 央行就必須付出相對的代價 這也是台灣大眾要負擔的成本 而<長期>貶低新台幣的成本累積 早晚有一天會壓垮央行 抑或是壓垮台灣的競爭力
貶低新台幣就好像貶低台灣的產品價格與勞動成本 最主要的就是犧牲台灣人民因勞動所得到的報酬
當然有人會認為筆者所言並不一定會發生 或舉出貶低新台幣的好處
沒錯 這些後遺症並不一定會發生 但事實上在台灣就有這些問題發生 有時候很難從匯率去比較 但如果從購買力指數 或大學生與專業人員的國際薪資水平 就可看出來台灣有很嚴重的問題所在 台灣的人才外流與外留已不是新鮮事 人才的流動可不是政府能夠控制的
醒醒吧 央行!
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