How to calculate odds for back door draws?

[Marcotte]

Situation:
I have A /images/graemlins/diamond.gif 3 /images/graemlins/diamond.gif
Flop: K /images/graemlins/diamond.gif Q /images/graemlins/club.gif 2 /images/graemlins/heart.gif

I have a backdoor flush and two backdoor straight draws, plus an overcard with no kicker. What odds do I need to call a bet on the flop?

More generally, how do you calculate odds/chances to make hand for backdoor draws?

[meep_42]

I know part of this would depend on the pre-flop action and how many people are in the pot (chances your A is outkicked)

-d

[fnord_too]

[ QUOTE ]
Situation:
I have A /images/graemlins/diamond.gif 3 /images/graemlins/diamond.gif
Flop: K /images/graemlins/diamond.gif Q /images/graemlins/club.gif 2 /images/graemlins/heart.gif

I have a backdoor flush and two backdoor straight draws, plus an overcard with no kicker. What odds do I need to call a bet on the flop?

More generally, how do you calculate odds/chances to make hand for backdoor draws?

[/ QUOTE ]

Let me answer for just the flush first:
You need two running diamonds, there are 10 diamonds left in the deck and 47 unknown cards, so you have a 10/47 chance of hitting the first diamond, if you do you have a 9/46 chance of hitting the second, for a total of
(10/47)(9/46) = ~0.042, or a little better than 1 in 25.

Computing for both the straight and the flush is slightly more difficult since the straight and flush cards have some overlap.

There are 16 ways to make the first part of the draw, 3 non diamond 4's, 3 non diamond 5's, and 8 diamonds not the 5 or 4, and two diamonds that give you two ways to hit. There are three types of good possible turns, and we need to see how each of these plays out:

Just the straight:
(6/47)(4/46) = ~ .011 (though there are only 6 non diamond 4 and 5's, the compling rank can be a diamond on the end.)

Just the flush:
(8/47)(9/46) = ~ .033 (note 9 diamonds complete the flush thought only 8 got you into this scenario).

Straight flush possible on turn:
(2/47)(12/46) = ~.011

Add these together and you get just 0ver 5.5%

Another approach:
You can count the number of two card hands that make your straight and/or flush and divide that by the total number of 2 card hands. The total is simply 47 choose 2, or (47*46)/2 = 1081.

You have 10 choose 2 ways of making the flush, that is 45.
You have 4*4 ways of making the straight, that is 16.
the 4 /images/graemlins/diamond.gif5 /images/graemlins/diamond.gif is counted in each of these, so we need to subtract one for the double counting. That gives us
60/1081 = just over 5.5%. The number is identical to the one above, which is reassuring /images/graemlins/smile.gif

[fnord_too]

Your question as to what odds do you need to call is more subtle. As one poster observed, you can hit your ace and still easily lose. Also you could hit your straight and tie or lose, though this is pretty unlikely. If you hit your flush (4.4% chance) however, you will definitely win if the board does not pair (barring someone hitting the runner runner straight flush). Even if the board pairs, your flush is still good most of the time. More possibilities include hitting running 3's or a running A3 which you give you a win, and runnig Aces probably gives you a win or a split with all the aces that outkick you without a K, Q, or 2, (providing trip aces isn't beaten by a boat or straight elsewhere).

You really have to look at the implied odds here. You know on the turn if your hand is done for or not, so if the turn is a blank you are only putting money in on the flop. You need to know how much is in the pot, how much you are going to likely have to spend to see the turn and the river if the turn gives you outs. Also, you need to know about how many bets you can extract from your opponent if you hit, and the chance and cost of hitting and still losing. Really, in this hand, the straight outs are pretty bad unless you catch the 3 /images/graemlins/diamond.gif4 /images/graemlins/diamond.gif on the turn, because you will only have 4 outs to make your straight, that leaves you worse than an 11-1 dog.

In these situations I think it is best to use simple heuristics: If you can take a card off for one small bet or better yet free, call if the pot is large and multi-way. If you pick up your flush draw, continue if you are getting good odds (usually that means calling one bet if you don't forsee a raise behind you).

The combinatorics are staggering in poker. I did not set out with the intent of naming so many possibilities, but as I type I keep thinking "Well, that's not 100% true becuase of this special case..."

[Marcotte]

Thanks fnord. That makes sense.

So, adding in the high straight draw (running JT), I get

Low Straight only
6/47 * 4/46 = ~1.11%
High Straight only
6/47 * 4/46 = ~1.11%
Flush only
6/47 * 9*46 = ~2.5%
(2, 6, 7, 8, 9, Qd for the first card, any d for the river)
Low and/or Flush
2/47 * 12/46 = ~1.11%
(4d or 5d followed by 9 diamonds or 3 non diamond 4s or 5s)
Hi and/or Flush
2/47 * 12/46 = ~1.11%

for a total of ~6.94% or 13.4:1 - this math look right?
So it seems that even with the best backdoor draws (without a pair), you'd want to fold unless the pot was raised with at least 6 callers, or three-bet with at least 4 callers. Need to learn to lay these down /images/graemlins/smirk.gif.

[Louie Landale]

I didn't cruitinize the math in the other posts but it looks good. However, they are showing you the chances or odds of making the hand. Since most of these draws will require you to invest another bet on the turn while you still just have a draw, your "pot odds" are inferior to a straight up draw: if you make your flush DRAW on the turn you'll have to invest more to keep drawing (about a quarter of the time it costs another 1bb).

So if its true and you are a 18:1 dog to make a straight or flush, you need MORE than 18:1 odds (counting implied odds) to call since the actual cost of your draw is MORE than the current one bet on the flop.

Did some fuzzy math on this a couple years ago and concluded that a back-door flush draw, nominally 23:1 against and about the equivalent of a 2-card out draw, is really only "worth" about 1.5 outs. You bad back-door straight odds are "worth" about .5 outs each. So totaling your 3 back-door draws together you get 2.5 outs; a little better than your 3-card out Ace draw, which will often lose when you make it.

These backdoor draws only add when you have some other way of winning, such as a weak over-card inside straight draw. The straight is worth ..err.. is 4 outs; the back-door flush 1.5, and your pair outs lets say worth about 2.5 (out of 6). That makes this hand "worth" about 8 outs.

Realistically, use your normal outs to figure if its worth it. If its close then let the presense of back-door draws decide.

- Louie