Help with calculations and notations

[gergery]

I’m trying to figure out what the math is and what the notations are for calculating various probabilities.

For example, I’m dealt 7h 6h. What are the chances I’ll get a flushdraw on the flop?
I know its about 11.5%, but how exactly do you calculate it, why multiply by those numbers, and how do you write it mathematically?

I seem to remember something like

11/50 for one heart * 10/49 for 2nd heart * 39/48 for non-heart * 3! / 2! Because your picking 3 cards but must have 2 of them be the ones you want

How would you write out the math if you’re dealt 76 preflop and want to know the odds you’ll flop an OESD?

[AncientPC]

This thread (http://forumserver.twoplustwo.com/showflat.php?Cat=&Number=1070814&page=0&view=colla psed&sb=5&o=14&fpart=1#1070814) might help.

[AceKQJT]

Holding 2 flush cards in your hand, I calculate the following flops:

3-Flush = 41.59% = 1 in 2.4
4-Flush = 10.94% = 1 in 9
Flush = 2.53% = 1 in 40

--Casey

--------EDIT---------------THE MATH-----------
To flop a 3-Flush =3*(11/50)*(39/49)*(38/48)
To Flop a 4-Flush =3*(11/50)*(10/49)*(39/48)
To flop a Flush 3*(11/50)*(10/49)*(9/48)

[AceKQJT]

OESD calculations are a little more complex. This is what I have come up with, and I think it is correct. Maybe one of the gurus can point out any mistakes I have made.

If we are holding 2 cards that can flop an OESD with 3 combinations of 3 different cards (such as 6-7 vs Q-J or 3-4) then we use the following (direct Excel formula):

3*(((COMBIN(8,2) - 2*COMBIN(4,2)) * 34) + (4*COMBIN(4,2)) + (4*COMBIN(4,2)))/COMBIN(50,3) = 9.06%

The breakdown of the above formula is (assuming we are holding 6-7):

We have to determine how many ways we can flop 2 cards to our open-ender (for instance, flop an 8-9-x when we hold 6-7). That is, how many 2-card combinations in 8 cards ---c(8,2) = 28. Problem is, the result includes 8-8 and 9-9 flops, which are no good to our OESD. there are c(4,2) = 6 ways to flop 8-8 and c(4,2) = 6 ways to flop 9-9, so we subtract those 12 ways from 28 to get 16 ways to flop 8-9.
{ c(8,2) - 2 * c(4,2) }

We haven't yet accounted for the 3rd flop card, which we do not want to make our straight. So 8-9 can make 16 combinations which can combine with 52-(2 in your hand)-(4 Tens)-(4 Fives)-(4 Sixes)-(4 Sevens) = 34 other cards. When we multiply that out, we get 16*34= 544 combinations of flops where x doesn't make us a straight and where x isn't another 8 or 9 (making an 8-8-9 or 8-9-9 board. Now we add those possibilities into the mix...

8-8-9 makes 4 * c(4,2) = 24 combonations, and 8-9-9 makes another 4 * c(4,2) = 24 combinations, so 48 total combinations of those hands.

We add 48 + 544 and get 592 possible OESD combinations using 8's and 9's.

{ (((c(8,2) - 2 * c(4,2)) * 34) + (4 * c(4,2)) + (4 * c(4,2))) }

So our OESD flop using 8 & 9 makes up 592 combinations, but OESD flops containing 4 & 5 and 8 & 5 also give us 592 combinations each, so we multiply 592 by 3 to get total 3-card combinations that give us an OESD...592 * 3 = 1776

now we simply divide that by the total enumerated boards c(50,3) = 19,600

that gives us 1776 / 19,600 = 0.0906 or 9.06%

Holding cards like 3-4 changes things, because we have only 2 combinations of 3 different cards (5 & 2, 5 & 6). Remember, A & 2 gives us a one-way draw.

Instead of multiplying the entire numerator (592) by 3, we multiply it by 2. That gives us:

2*(((COMBIN(8,2) - 2*COMBIN(4,2)) * 34) + (4*COMBIN(4,2)) + (4*COMBIN(4,2)))/COMBIN(50,3) = 6.04%


Wow...Hope someone checks my math

--Casey

[AceKQJT]

On calculating flopping a flush, I mistakenly multiplied by 3, when there is only 1 way to flop a flush. SO...

To flop a flush = (11/50)*(10/49)*(9/48) = 0.84% = 1 in 118.8

--Casey

[Indiana]

You got 67h? Then the flop is a choosing of 3 cards from 50 in the deck. You have 11 hearts left and 39 non flush cards. So the prob=C(11,2)*C(39,1)/C(50,3) = .11 or 11%...which gives us odds of .89/.11= 8 to 1...

What is an OESD?

Indiana

[Edge34]

OESD = Open End Straight Draw.