Re: HE Odds
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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
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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.
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B. How often you have a 4 flush on the turn if you start with 2 suited
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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.
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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.
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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.
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