Odds on runner-runner flush OR RR-straight

[psychprof]

I tried to do this myself, but would appreciate some verification. This is my first attempt at probability calculations.

Let's say I have Jh 5h and flop is

4h Js 3d

The flop gives me a runner-runner flush draw and a runner-runner straight draw. I have calculated the odds of making either the flush OR the straight, and I would appreciate it if you would check my calculations.

BTW, as my name indicates, I am a psychology professor so I have some experience with probability. However, my experience is mainly in the area of hypothesis testing with the normal curve, so standard probability calculations like the ones below are kinda rusty for me. I appreciate your time and corrections.

OK, I start with the easy one.

Probability of a runner runner flush (RRF)

P(heart on turn) = 10/47
P(heart on river given heart on turn) = 9/46
P(RRF)= 10/47 * 9/46 = .0416

*I forget the symbol for "given".

Probability of a runner runner straight (RRS)

P(A, 2, 6, or 7 on turn) = 16/47

If the turn gives me an open-ended straight draw (let's say a 2) then...
P(A or 6 on river) = 8/46
So the probability of completing an RR-OE straight is
P(RR-OE-S) = 16/47 * 8/46 = .059

If the turn gives me a gutshot straight draw (let's say a 7) then...
P(6 on river) = 4/46
So the probability of completing a RR-GS straight is
P(RR-GS-S) = 16/47 * 4/46 = .030

Probability of completing a RR-Flush OR a RR-GS-Straight
P(RR-F OR RR-GS-S) = .0416 + .030 = .0716 or a 7.2% chance

Probability of competing a RR-Flush OR a RR-OE-Straight
P(RR-F OR RR-OE-S) = .0416 + .059 = .10 or a 10% chance

In conclusion, the probability of completing a RR-flush OR a RR-straight are somewhere between 7 and 10 percent. Using the more generous 10%, this means I'm a 9:1 dog to improve the hand (to a straight or flush) and so I need better than 9:1 pot odds to justifying staying in.

So, if there's a $2 bet to me and the pot already holds $20 I have the odds to call (10:1).


OK, that took me a while, and I wouldn't be surprised if I were wrong in several places. I sincerely appreciate the time it takes you to reply and thank you for your help.
PsychProf

PS. Maybe I should have picked an easier problem to tackle as my first probability calculation in 10 years. LOL

[Bozeman]

It seems that you made a bunch of mistakes.

First, you are counting str8 flush cards twice. Second, you need to separate the OE's from the GS's, you doubled your turn cards for each.

I don't guarantee my results either, but I get:

P(flush)=10/47*9/46=4.16%

P(OE str8, no flush card on 4th)=6/47*8/46=2.22%

P(OE str8, flush card on 4th)=2/47*6/46=0.56%

P(GS str8, no flush card on 4th)=6/47*4/4=1.11%

P(GS str8, flush card on 4th)=2/47*3/46=0.28%

So P(str8 or flush)=8.33% (11:1)

Interestingly, your backdoor str8, no flush chances exactly equal your flush chances.

[Copernicus]

Even though you didnt guarantee your results, they are correct.

[GuyOnTilt]

[ QUOTE ]
Using the more generous 10%, this means I'm a 9:1 dog to improve the hand (to a straight or flush) and so I need better than 9:1 pot odds to justifying staying in.

So, if there's a $2 bet to me and the pot already holds $20 I have the odds to call (10:1).

[/ QUOTE ]

This is incorrect. In addition to the flop call, you will also usually have to call one or two bets on the turn, meaning you will need MUCH better than 10:1 on the flop. This is why runner-runner draws are virtually never worth drawing to alone.

[elwoodblues]

Don't forget that he also has top pair - weak kicker. Couple that with the straight and flush potential, I personally think a call could be easily justified.

[psychprof]

Thanks Bozeman for the corrections (and to Copernicus for the verification). I understand everything you did and why you did it. I think I was just trying to take a shortcut on the straight draw calculations. Precision is definitely better!

Thanks GuyonTilt for your comment as well. If I remember my play right (this was an actual hand for me last week) I did fold on the flop. My comment that you quoted was included more for the sake of checking my understanding of how pot odds and hand odds interact to determine optimum play. It helps to know that I should consider future streets in my odds calculations.

Elwoodblues -- You may be right, but I think with enough people staying in, the weak kicker ruins my top pair, and there are several overcards that could come as well. With this flop I felt like my only chances of winning were the straight or flush, and even the Jack high flush is suspect.

To be honest, it was one of those hands that I really wanted to see the turn card, but intuition, reason, or wimpiness won over and I folded.

(Note that I did not say "odds calculations" caused me to fold, because there is no way I could figure the odds on this hand in the 30 seconds I have to respond [img]/images/graemlins/confused.gif[/img]. I truly don't know how you guys do it on the fly. It's very impressive!)

[Copernicus]

Very little odds calculating is done on the fly. The more standard drawing odds such as your question are memorized, and less common ones are usually estimated. I tend to use (48- number of outs that are nearly certain winners * cards to come)/(number of outs.....to come). The value gained from precision is minimal compared to what is gained by spending the same amount of time reading your opponents hand, imo.

[GuyOnTilt]

[ QUOTE ]
Don't forget that he also has top pair - weak kicker.

[/ QUOTE ]

That nothing to do with him doing bad math. I was merely pointing out his errors in calculation, which are very significant.