HE Odds

[Shandrax]

Everyone knows the odds for the flop, but does anyone know:

A. How often you have a set on the turn(!) if you start with a pair
B. How often you have a 4 flush on the turn if you start with 2 suited
C. This is a bit more tricky because of the gaps, let's say how often you have a 4 straight on the turn if you start with J-T.

[Blitzwing]

A. Assuming you didnt have a set on the flop there are still 2 outs to make the set. With the flop on the table there are 47 unseen cards. 2/47 = 4,3%

B. A 4 flush on the turn with suited in the hole means you hit only 1 suit on the flop. 1 suit on the flop means 11 out on the flop and 10 outs on the turn.

39/50 * 38/49 * 37/48 = 46,6% chance not hitting a suit on the flop means a 53,4% of hitting a suit on the flop.

39/50 * 38/49 * 37/48 * 35/48 = 65,3% chance not hitting 4 flush on the turn means a 34,7% chance of hitting e flush on the turn.

C. Im still working on this 1!

***note***
If im incorrect which is possible then it would be great is some1 could correct me. And if possible give the right solution in the way I calculated it.
***note***

[Shandrax]

Someone send the batsign to BruceZ.

[BruceZ]

[ QUOTE ]
Everyone knows the odds for the flop, but does anyone know:

A. How often you have a set on the turn(!) if you start with a pair

[/ QUOTE ]

1 - C(48,4)/C(50,4) =~ 15.5%

OR

1 - (48/50 * 47/49 * 46/48 * 45/47) =~ 15.5%

This includes full houses and quads.


[ QUOTE ]
B. How often you have a 4 flush on the turn if you start with 2 suited

[/ QUOTE ]

C(11,2)*C(39,2) / C(50,4) =~ 17.7%

OR

11/50 * 10/49 * 39/48 * 38/47 * C(4,2) =~ 17.7%

where we multiply by C(4,2) = 6 positions for the 2 flush cards.


[ QUOTE ]
C. This is a bit more tricky because of the gaps, let's say how often you have a 4 straight on the turn if you start with J-T.

[/ QUOTE ]

I'll assume that you mean 8-out straight draws only, not gut shots, but I will include double gut shots. For simplicity, this will include 4-straights that are also made flushes, since trying to eliminate these is messy, the difference is small, and you are usually happy to make a flush in addition to a 4-straight. This will also include 4-straights where the board pairs, or that make flush draws, but not made straights.

[ 3*(16*34*30/2 + 2*6*4*34 + 6*6) + 2*(4*4*4*30 + 3*6*4*4)] / C(50,4)

=~ 14.7%.

There are 3 2-card combinations that you can flop to make an open-ended straight draw (e.g., for JT they are 98, Q9, and KQ). There are 16 ways to make each of these, and there are 48-8-6 = 34 remaining cards that don't complete the straight (8) and don't pair the board (6), so there are 34*30/2 ways to choose the other 2 board cards without pairing the board or completing the straight. The 2nd term is for the paired boards with a single pair. There are 2 ranks that can pair, times C(4,2) = 6 ways to make the pair, times 4 ways to choose the non-paired card, times 34 ways to choose the last card without pairing the board or completing the straight. The 3rd term is for pairing both board cards (e.g. 8989), and there are C(4,2) = 6 ways to choose each pair. The next term is for the double gut shots. There are 2 of these ignoring suit (e.g. for JT they are K97 and AQ8), and there are 4 ways to choose each of the 3 cards, times 47-8-9 = 30 cards that don't complete the straight (8) or pair the board (3*3=9). The last term is for the double gut shots that pair the board. There are 3 cards that can pair, times C(4,2) = 6 ways to choose the pair, times 4 ways to choose each of the other 2 cards.

[BruceZ]

[ QUOTE ]
39/50 * 38/49 * 37/48 * 35/48 = 65,3% chance not hitting 4 flush on the turn means a 34,7% chance of hitting <font color="red">4</font> flush on the turn. <font color="red">(Changed 'e' to '4' - BZ)</font>

[/ QUOTE ]

You meant 1 - (39/50 * 38/49 * 37/48 * 36/47) = 65,3%, BUT 1 - P(no flush cards on board at turn) does not equal the probability of a 4-flush on the turn. This is just the probability of at least 1 flush card on board at the turn.

Not every problem should be done with the 1 minus the probability trick. See my post in this thread for the calculation.

[Shandrax]

BruceZ, you are awesome. Thanks a ton!