Odds of open-ended straight improving

[g4p2005]

I have a question for you, math lovers:
If I have an open-ended straight after flop, what are the odds of
hitting the fifth card by river? I found 41% in an odds section of a
website, but it seems too high for me. I tried to figure it out by
myself, please correct me:
For one of the two expected cards:
4/47 odds of hitting on turn
4/46 odds of hitting on river.
Then, odds of hitting on turn or river are 4/47+4/46 - 16/(47*46) =
16.5%
Then, for both two expected cards, the odds will be 2*16.5%=33%.

[Jacob_Gilliam]

Your math is right. This is an easy question so why ask....

[gamble4pro]

First of all, 41% is incorrect, indeed.
Although your calculations are mostly incorrect, you have reached a result pretty close to real odds.
Your event "hitting on river" is ambiguously defined. You should consider "hitting on river if not hit on turn". So, even if applied formula is correct, the entire calculation becomes incorrect.
Here is the correct result, obtained by using combinations:
Let C and D be the expected cards (as value). 4 outs each.
The favorable combinations of turn+river cards are:
(Cx) - 4*(47-8)
(Dx) - 4*(47-8)
(CD) - 4*4
(CC) - C(4,2)
(DD) - C(4,2)
(x different from C and D)
Totally, we have 340 favorable combinations from C(47,2)=1081 possible,
so the odds of hitting the straight by river are 31.45%.

[LetYouDown]

I come up with the same number. Random question. When I am calculating odds on a hand, I invariably return to the combinations method. For this example I figured the # of combinations for:

A-A
B-B
A-B
A-X (not B)
B-X (not A)

1081 - sum(above)

I can do this relatively quickly in my head (I've memorized enough flop/turn combinations) but is there a better way? This method seems pretty solid to me. I've never been a fan of the "take 1 - the odds that you won't hit" approach...but perhaps I'm missing out.

The initial calculation could intuitively be proven wrong based on the fact that a flush is ~35% (9 outs) and a an OESD must be less (8 outs).

[LetYouDown]

In case I wasn't clear...the method I'm referring to is as follows.

There's 1081 possible turn/river combinations in this scenario, as we well know. If you're going to miss your straight, there's 39 cards that can come on the turn/river. 39 C 2 turns out to be 741...being the odds that you don't hit. 1081-741/1081 = 340/1081 = 31.45%. This method seems easier, but half the time I end up doing it the "hard" way.

[olavfo]

Straightforward calculation without using combinations:

You have 8 outs. Before the turn there are 47 unknown cards. 39 of them won't help you, so you have a 39/47 chance of missing on the turn

If you miss on the turn, there are now 46 unknown cards before the river, and 38 of them won't help you. You now have a 38/46 chance of also missing on the river.

The chance of missing on both the turn and river is:

P(miss) = (39/47)*(38/46) = 1482/2161 = 0.6855

The chance of improving on either turn or river is therefore:

P(improve) = 1 - P(miss) = 1 - 0.6855 = 0.3145 = 31.45%

Use the same math for flushes (with 9 outs instead of 8).

olavfo

[Jedi Flopper]

Why make it so hard? Use the shortcut. (# of outs) * (# of cards to come) * 2 will give you a very accurate estimate of the actual odds.

Open ended straight on the flop? 8 * 2 * 2 = 32%

Four flush on the turn? 9 * 1 * 2 = 18%

2 overs and a str flush draw on the flop 21 * 2 * 2 = 84% chance of improving to at least top pair by the river.

Gotta love simple.

[olavfo]

[ QUOTE ]
2 overs and a str flush draw on the flop 21 * 2 * 2 = 84% chance of improving to at least top pair by the river.

Gotta love simple.

[/ QUOTE ]
Too simple, since the correct probability with 21 outs is 70%, not 84%. The 2-4 rule isn't very accurate when the number of outs is large, but it works reasonably well for 10 outs or less.

olavfo

[Mike Haven]

i notice you, g4p2005, asked exactly the same question in thehendonmob.com forum and gamble4pro gave exactly the same answer, both at almost exactly the same times

is there some hidden agenda here by any chance? you wouldn't be the same poster about to spam something or other would you?

edit: yes, i see your game now, having read this thread

you should note that twoplustwo doesn't like spammers