questions regarding odds

I would appreciate it if someone would help me with a couple of questions regarding some odds:


1. What are the odds of flopping a set when you hold a pair?

2. What are the odds of flopping a four flush when you are suited?

3. What are the odds that you will flop an open ended straight draw when your hand is connected?

4. What are the odds that you will flop and open ended straight draw OR flush draw if your hand is suited and connected?


I think that 3 and 4 might depend on how close your cards are to the extremes, if so just assume they are 89 or somewhere in the middle.


Any help with any or all of these questions would be appreciated. I would also love to know how these odds are calculated so I can answer my own questions going forward but I understand if it is too difficult to explain.


Thanks

Actually I want the odds that you flop


1.a set or quads

2.a flush draw or a flush

3.a straight draw or a straight

4.a straight draw, flush draw, straight, or flush


sorry for the vagueness in my prior post and thanks in advance for any help.

1) Its easy to find the probability you don't flop a set. Its

= 48/50 (47/49)( 46/48). So its one minus this.


2) There are 11C2(39) such flops. SO the probability you get one of these flops is 11C2(39)/50C3. There are also 11C3 such flops in which you flop a flush.


3) Lets assume you have JT. There 5 basic flops that make you open ended or a double belly buster (and not a straight) KQx, Q9x, 98x, AQ8 , K97. The number of such boards which are not paired is 16(38)+ 16(34) +16(34) +(16)(34)+ 2(64).

There are also 4(64) ways to flop a straight. There are also 3(48) ways to flop an open ended straight draw when the board is paired.

Suspicious did the hard part (finding the number of flops which make your hand), but he didn't finish by dividing by the total number of flops (which is C(50,2)=19600, given your hole cards) to get the probability. Since I'm not sure if you know to do this, I'll finish it for you. By the way, C(n,k) denotes the number of ways to choose k objects from a set of n objects. This is calculated as n! / (k! * (n-k)!).


1) There are C(48,3) or 17296 flops which don't contain a card of your pair. This means 19600-17296=2304 flops contain at least one card of your pair. The probability of such a flop is 2304/19600=0.1176 or 11.76%.


2) There are C(11,2)*C(39,1)=2145 flops with two cards of your suit, and C(11,3)=165 flops with 3 cards of your suit. So the total flops with a flush or flush draw are 2145+165=2310, making the probability 2310/19600=0.1179 or 11.79%.


3) As suspicious outlined, there are 3 different ways to flop open ended, and 2 ways to flop a double gutshot (assuming you hold connectors 54 up to JT). There are C(4,1)*C(4,1)*C(38,1)=608 ways to flop a specific open ended straight draw with no pair on the board, multiplied by the 3 ways to be open ended is 1824 such flops -- this includes those times you flop open ended while pairing one of your hole cards. There are C(4,2)*C(4,1)*2=48 ways (this is multipled by two because either card could be paired) to flop a specific open ended straight draw where the board is paired, multiplied by the 3 ways to be open ended is 144 such flops. There are C(4,1)*C(4,1)*C(4,1)=64 ways to flop a specific double gutshot, multiplied by the two ways to have a double gutshot is 128 such flops. There are C(4,1)*C(4,1)*C(4,1)=64 ways to flop a specific straight, multiplied by the 4 ways to flop a straight is 256 such flops. So there are 1824+144+128+256=2352 total such flops, and the probability is 2352/19600=0.12 or 12%.


4) To do this we are going to add the number of flops from 3 and 4 above, but we must subtract out those flops which we counted twice because they contained both a straight (or a draw to it) and a flush (or a draw to it). For each open ended straight draw with no pair on board, there are C(1,1)*C(1,1)*C(9,1) or 9 ways to have a flush, C(1,1)*C(1,1)*C(29,1) or 29 ways to have an open ended straight flush, and C(9,1)*C(3,1)*C(1,1)*2 (multiplied by two because either straight card could be offsuit) or 54 ways to be open ended with a flush draw but no straight flush draw -- multiplied by the 3 ways to be open ended is 276 such flops. For each open ended straight draw with a pair on board, there are C(1,1)*C(1,1)*C(6,1) or 6 ways to have a flush draw as well, multiplied by the 3 ways to be open ended is 18 such flops. For each double gutshot, there is C(1,1)*C(1,1)*C(1,1)=1 way to flop a flush plus the C(3,1)*C(1,1)*C(1,1)*3=9 ways (multiplied by 3 because any of the 3 cards could be the offsuit one) to have a flush draw, multiplied by the 2 ways to have a double gutshot is 20 such flops. For each flopped straight, there is C(1,1)*C(1,1)*C(1,1)=1 way it could be a straight flush plus the C(3,1)*C(1,1)*C(1,1)*3=9 ways (again, multiplied by 3 because any of the 3 cards could be the offsuit one) it could contain a flush draw, multiplied by the four ways to flop a straight, is 40 such flops. So there are 276+18+20+40=354 straight draw or made straight flops which also contain a flush draw or a made flush, which means there are 2310+2352-354=4308 flops which help your suited connector 54-JT, making the probability of such a flop 4308/19600=0.2198 or 21.98%.

I made a few small errors in 3 and 4. They should read:


3) As suspicious outlined, there are 3 different ways to flop open ended, and 2 ways to flop a double gutshot (assuming you hold connectors 54 up to JT). There are C(4,1)*C(4,1)*C(34,1)=544 ways to flop a specific open ended straight draw with no pair on the board, multiplied by the 3 ways to be open ended is 1632 such flops -- this includes those times you flop open ended while pairing one of your hole cards. There are C(4,2)*C(4,1)*2=48 ways (this is multipled by two because either card could be paired) to flop a specific open ended straight draw where the board is paired, multiplied by the 3 ways to be open ended is 144 such flops. There are C(4,1)*C(4,1)*C(4,1)=64 ways to flop a specific double gutshot, multiplied by the two ways to have a double gutshot is 128 such flops. There are C(4,1)*C(4,1)*C(4,1)=64 ways to flop a specific straight, multiplied by the 4 ways to flop a straight is 256 such flops. So there are 1632+144+128+256=2160 total such flops, and the probability is 2160/19600=0.1102 or 11.02%.


4) To do this we are going to add the number of flops from 3 and 4 above, but we must subtract out those flops which we counted twice because they contained both a straight (or a draw to it) and a flush (or a draw to it). For each open ended straight draw with no pair on board, there are C(1,1)*C(1,1)*C(7,1)=7 ways to have a flush, plus the C(1,1)*C(1,1)*C(27,1)=27 ways to have an open ended straight flush, plus the C(7,1)*C(3,1)*C(1,1)*2=42 ways (multiplied by two because either straight card could be offsuit) to be open ended with a flush draw but no straight flush draw -- multiplied by the 3 ways to be open ended is 228 such flops. For each open ended straight draw with a pair on board, there are C(1,1)*C(1,1)*C(6,1)=6 ways to have a flush draw as well, multiplied by the 3 ways to be open ended with a pair on board is 18 such flops. For each double gutshot, there is C(1,1)*C(1,1)*C(1,1)=1 way to flop a flush plus the C(3,1)*C(1,1)*C(1,1)*3=9 ways (multiplied by 3 because any of the 3 cards could be the offsuit one) to have a flush draw, multiplied by the 2 ways to have a double gutshot is 20 such flops. For each flopped straight, there is C(1,1)*C(1,1)*C(1,1)=1 way it could be a straight flush plus the C(3,1)*C(1,1)*C(1,1)*3=9 ways (again, multiplied by 3 because any of the 3 cards could be the offsuit one) it could contain a flush draw, multiplied by the four ways to flop a straight, is 40 such flops. So there are 228+18+20+40=306 straight draw or made straight flops which also contain a flush draw or a made flush, which means there are 2310+2160-306=4164 flops which help your suited connector 54-JT, making the probability of such a flop 4164/19600=0.2124 or 21.24%.