Cyrus
05-20-2003, 05:44 AM
We hold JhTh and we want the probability of flopping a 4-Flush draw, without flopping a Straight, and without flopping a 4-Flush AND a Straight draw :
There are (13 hearts in all - 2 hearts in our hand)=11 hearts remaining in the deck. Since C(11,2)=55, there can be 55 2-card combos made from 11 hearts. There are (52 -2 cards in our hand -11 other hearts) = 39 other cards which can act as a 3rd card to complete the flop. So there are 55*39=2145 possible 2-heart flops in all.
We must sanitize the above number of 2145 combos by taking out all the unwanted flops. When we sanitize it, we will divide the sanitized number by the number of all possible 3-card flops and we will obtain the desired probability.
The sanitizing process involves taking out
- the 4-Flush and Straight-making flop combinations,
- the 4-Flush and open-ended Straight-draw flop combinations,
- the 4-Flush and Double-Belly Buster Straight-draw flop combinations, and
- the 4-Flush and Inside Straight-draw flop combinations.
The first act is the toughest. We got to identify the NON-Straight-making flops and deduct them from the total number of the total 2-heart flops in order to get the 4-Flush and Straight-making flop combinations.
We take out the combos whereby the 3rd card makes a Straight. Each flop's non-suited card is inside the parenthesis: 7h8h(9); 7h9h(8); 8h9h(7); 8h9h(Q); 8hQh(9); 9hQh(8); 9hQh(K); 9hKh(Q); QhKh(9); QhKh(A); AhQh(K); AhKh(Q).
The number of available cards from which we will be making up every time the combinations of flops is (52 cards - 2 cards in our hand - 2 cards outside the parenthesis every time)= 48. If we take out the hearts, the remaining hearts are (13 hearts in all - 2 hearts in our hand - 2 hearts in each of the above combos)= 9 hearts remaining every time.
There are (48 - 3 Nines - 9 remaining hearts)=36 cards whereby each can complete the [78X] flop to produce a 3-card flop. This makes for 36 flops. Each of these flops does not flop us a Straight. So, the number of 3-card flops, which every 2-card combo from the above detailed combos, can produce without making a Straight, is
For 78 : 48 - 3 Nines - 9 remaining hearts = 36
For 79 : 48 - 3 Eights - 9 remaining hearts = 36
For 89 : 48 - 3 Sevens - 3 Queens - 9 remaining hearts = 33
For 8Q : 48 - 3 Nines - 9 remaining hearts = 36
For 9Q : 48 - 3 Eights - 3 Kings - 9 remaining hearts = 33
For 9K : 48 - 3 Queens - 9 remaining hearts = 36
For QK : 48 - 3 Nines - 3 Aces - 9 remaining hearts = 33
For AQ : 48 - 3 Kings - 9 remaining hearts = 36
For AK : 48 - 3 Queens - 9 remaining hearts = 36
36+36+33+36+33+36+33+36+36= 315
The other 2-heart, 2-card combinations that are in no danger of producing a Straight are 55combos - 9 combos of 78/89/etc = 46combos. Each of those safe combos can combine with the (52cards - 2hearts in our hand - 2hearts in the flop - 9hearts remaining) = 39 non-heart cards, to give 46*39= 1794 combinations which don not produce a Straight.
The total flop combos that do not produce a Straight are 315+1794= 2109. Therefore, the total flops that do produce a Straight (along with a 4-Flush draw) are 2145-2109= 36.
Amongst the 2109 combos that do not produce a Straight right away (see previous), there are some combos that produce a 4-Flush and an open-ended Straight-draw flop. Those possible combos are
89X (where X is not 7 or Q)
9QX (where X is not 8 or K)
QKX (where X is not 9 or A)
For the X89, we can have three cases :
(a) 8h9h and Xd or Xc or Xs,
where X is d,c,s of 2,3,4,5,6,8,9,T,J,K,A=11cards
So 3*11=33cards which can be X, making 33 combos for (a)
(b) 8hXh and 9d or 9c or 9s,
Now X must be a heart from 2,3,4,5,6,K,A=7cards
So, the (1 card for 8h)*(7cards for X)*(3cards for 9 d/c/s)=21cards, making 21 combos for (b)
(c) 9hXh and 8d or 8c or 8s,
Again X must be a heart from 2,3,4,5,6,K,A=7cards
So, the (1 card for 9h)*(7cards for X)*(3cards for 8 d/c/s)=21cards, making 21 combos for (b)
The total of (a)+(b)+(c)= 33+21+21 = 75 flop combos that produce a 4-Flush and an open-ended 89X Straight-draw. The same calculation is done for 9QX and QKX so, there are 3*75= 225 flop combos that produce a 4-Flush and an open-ended Straight-draw.
Next, the Double-Belly-Buster Straight-draws : The possible DBB flops are
8QA and 79K. The first possible flop can be made C(3,2)= 3 ways cause there are 3 2-heart combinations made of 3 hearts. Each of those 2-heart combinations combines with the non-heart suits of the other, the third rank in the flop and gives 3*3 =9 DBB combinations with 2 hearts. Same is true for the second possible DBB flop, so 2*9 = 18 total 4-Flush and DBB Straight-draw flop combinations.
Finally, the Inside Straight draws, along with the 4-Flush. The possible Inside Straight-drawing 2-heart flops are found in 6 possible flop structures:
AKX
AQX
9KX
8QX
79X
78X
We have to take out the flops that produce a Straight ...straight away.
For the AKX case, the possible 2-heart flops are
(a) AhKh and Xd or Xs or Xc
(b) AhXh and Kd or Ks or Kc
(c) KhXh and Ad or As or Ac
For (a), X can be d,s,c of 2,3,4,5,6,7,8,9,T,J,K,A= 12 cards (the Q-rank is out)
so 3*12 = 36cards which can be X, making 36 combos for (a).
For (b), X must be a heart and K is d,s,c. X can be a heart from 2,3,4,5,6,7,8,9,K= 9cards (the Jh & Th we already hold in our hand, Ah is already in the flop and Qh flops a Straight) so
(1 card for Ah)*(9 cards for X)*(3 cards for K d/s/c) = 27 combos for (c).
For (c), X must be a heart and A is d,s,c. X can be a heart from 2,3,4,5,6,7,8,9,A= 9cards (the Jh & Th we already hold in our hand, Kh is already in the flop and Qh flops a Straight) so
(1 card for Kh)*(9 cards for X)*(3 cards for A d/s/c) = 27 combos for (c).
Therefore, there are 36+27+27 = 90 2-heart flops that give a 4-Flush draw and an Inside Straight draw for AKX. The same results we get for the other five structures, so we have 5*90=540 total 2-heart flop combinations that give a 4-Flush draw and an Inside Straight draw.
Summing up, we have
36 flop combinations that produce a 4-Flush draw and a Straight,
225 that produce a 4-Flush and open-ended Straight-draw
18 that produce 4-Flush and Double-Belly Buster Straight-draw flop, and finally
540 that produce 4-Flush and Inside Straight-draw flop combinations
The total is 36+225+18+540= 819, which we deduct from the (55*39)=2145 total possible 2-heart flops, to get 2145 - 819 = 1326 sanitized flops that produce ONLY a 4-Flush draw.
Therefore, the incredibly valuable /forums/images/icons/grin.gif probability of flopping only a 4-Flush draw, when holding JhTh, is 1326 divided by the total number of 2-card flops, ie C(50,2)=19600, so 1326/19600= <font color="red"> 7%</font color> or 6.765306%.
I would welcome any corrections or suggestions for short-cuts to the above process.
--Cyrus
There are (13 hearts in all - 2 hearts in our hand)=11 hearts remaining in the deck. Since C(11,2)=55, there can be 55 2-card combos made from 11 hearts. There are (52 -2 cards in our hand -11 other hearts) = 39 other cards which can act as a 3rd card to complete the flop. So there are 55*39=2145 possible 2-heart flops in all.
We must sanitize the above number of 2145 combos by taking out all the unwanted flops. When we sanitize it, we will divide the sanitized number by the number of all possible 3-card flops and we will obtain the desired probability.
The sanitizing process involves taking out
- the 4-Flush and Straight-making flop combinations,
- the 4-Flush and open-ended Straight-draw flop combinations,
- the 4-Flush and Double-Belly Buster Straight-draw flop combinations, and
- the 4-Flush and Inside Straight-draw flop combinations.
The first act is the toughest. We got to identify the NON-Straight-making flops and deduct them from the total number of the total 2-heart flops in order to get the 4-Flush and Straight-making flop combinations.
We take out the combos whereby the 3rd card makes a Straight. Each flop's non-suited card is inside the parenthesis: 7h8h(9); 7h9h(8); 8h9h(7); 8h9h(Q); 8hQh(9); 9hQh(8); 9hQh(K); 9hKh(Q); QhKh(9); QhKh(A); AhQh(K); AhKh(Q).
The number of available cards from which we will be making up every time the combinations of flops is (52 cards - 2 cards in our hand - 2 cards outside the parenthesis every time)= 48. If we take out the hearts, the remaining hearts are (13 hearts in all - 2 hearts in our hand - 2 hearts in each of the above combos)= 9 hearts remaining every time.
There are (48 - 3 Nines - 9 remaining hearts)=36 cards whereby each can complete the [78X] flop to produce a 3-card flop. This makes for 36 flops. Each of these flops does not flop us a Straight. So, the number of 3-card flops, which every 2-card combo from the above detailed combos, can produce without making a Straight, is
For 78 : 48 - 3 Nines - 9 remaining hearts = 36
For 79 : 48 - 3 Eights - 9 remaining hearts = 36
For 89 : 48 - 3 Sevens - 3 Queens - 9 remaining hearts = 33
For 8Q : 48 - 3 Nines - 9 remaining hearts = 36
For 9Q : 48 - 3 Eights - 3 Kings - 9 remaining hearts = 33
For 9K : 48 - 3 Queens - 9 remaining hearts = 36
For QK : 48 - 3 Nines - 3 Aces - 9 remaining hearts = 33
For AQ : 48 - 3 Kings - 9 remaining hearts = 36
For AK : 48 - 3 Queens - 9 remaining hearts = 36
36+36+33+36+33+36+33+36+36= 315
The other 2-heart, 2-card combinations that are in no danger of producing a Straight are 55combos - 9 combos of 78/89/etc = 46combos. Each of those safe combos can combine with the (52cards - 2hearts in our hand - 2hearts in the flop - 9hearts remaining) = 39 non-heart cards, to give 46*39= 1794 combinations which don not produce a Straight.
The total flop combos that do not produce a Straight are 315+1794= 2109. Therefore, the total flops that do produce a Straight (along with a 4-Flush draw) are 2145-2109= 36.
Amongst the 2109 combos that do not produce a Straight right away (see previous), there are some combos that produce a 4-Flush and an open-ended Straight-draw flop. Those possible combos are
89X (where X is not 7 or Q)
9QX (where X is not 8 or K)
QKX (where X is not 9 or A)
For the X89, we can have three cases :
(a) 8h9h and Xd or Xc or Xs,
where X is d,c,s of 2,3,4,5,6,8,9,T,J,K,A=11cards
So 3*11=33cards which can be X, making 33 combos for (a)
(b) 8hXh and 9d or 9c or 9s,
Now X must be a heart from 2,3,4,5,6,K,A=7cards
So, the (1 card for 8h)*(7cards for X)*(3cards for 9 d/c/s)=21cards, making 21 combos for (b)
(c) 9hXh and 8d or 8c or 8s,
Again X must be a heart from 2,3,4,5,6,K,A=7cards
So, the (1 card for 9h)*(7cards for X)*(3cards for 8 d/c/s)=21cards, making 21 combos for (b)
The total of (a)+(b)+(c)= 33+21+21 = 75 flop combos that produce a 4-Flush and an open-ended 89X Straight-draw. The same calculation is done for 9QX and QKX so, there are 3*75= 225 flop combos that produce a 4-Flush and an open-ended Straight-draw.
Next, the Double-Belly-Buster Straight-draws : The possible DBB flops are
8QA and 79K. The first possible flop can be made C(3,2)= 3 ways cause there are 3 2-heart combinations made of 3 hearts. Each of those 2-heart combinations combines with the non-heart suits of the other, the third rank in the flop and gives 3*3 =9 DBB combinations with 2 hearts. Same is true for the second possible DBB flop, so 2*9 = 18 total 4-Flush and DBB Straight-draw flop combinations.
Finally, the Inside Straight draws, along with the 4-Flush. The possible Inside Straight-drawing 2-heart flops are found in 6 possible flop structures:
AKX
AQX
9KX
8QX
79X
78X
We have to take out the flops that produce a Straight ...straight away.
For the AKX case, the possible 2-heart flops are
(a) AhKh and Xd or Xs or Xc
(b) AhXh and Kd or Ks or Kc
(c) KhXh and Ad or As or Ac
For (a), X can be d,s,c of 2,3,4,5,6,7,8,9,T,J,K,A= 12 cards (the Q-rank is out)
so 3*12 = 36cards which can be X, making 36 combos for (a).
For (b), X must be a heart and K is d,s,c. X can be a heart from 2,3,4,5,6,7,8,9,K= 9cards (the Jh & Th we already hold in our hand, Ah is already in the flop and Qh flops a Straight) so
(1 card for Ah)*(9 cards for X)*(3 cards for K d/s/c) = 27 combos for (c).
For (c), X must be a heart and A is d,s,c. X can be a heart from 2,3,4,5,6,7,8,9,A= 9cards (the Jh & Th we already hold in our hand, Kh is already in the flop and Qh flops a Straight) so
(1 card for Kh)*(9 cards for X)*(3 cards for A d/s/c) = 27 combos for (c).
Therefore, there are 36+27+27 = 90 2-heart flops that give a 4-Flush draw and an Inside Straight draw for AKX. The same results we get for the other five structures, so we have 5*90=540 total 2-heart flop combinations that give a 4-Flush draw and an Inside Straight draw.
Summing up, we have
36 flop combinations that produce a 4-Flush draw and a Straight,
225 that produce a 4-Flush and open-ended Straight-draw
18 that produce 4-Flush and Double-Belly Buster Straight-draw flop, and finally
540 that produce 4-Flush and Inside Straight-draw flop combinations
The total is 36+225+18+540= 819, which we deduct from the (55*39)=2145 total possible 2-heart flops, to get 2145 - 819 = 1326 sanitized flops that produce ONLY a 4-Flush draw.
Therefore, the incredibly valuable /forums/images/icons/grin.gif probability of flopping only a 4-Flush draw, when holding JhTh, is 1326 divided by the total number of 2-card flops, ie C(50,2)=19600, so 1326/19600= <font color="red"> 7%</font color> or 6.765306%.
I would welcome any corrections or suggestions for short-cuts to the above process.
--Cyrus