Flush Flops

[MrDannimal]

I have a friend who is convinced that there are "too many" flops/boards at UltimateBet that make flushes (meaning too many boards with 3 or 4 card flushes).

I keep trying to tell him that:

a) No.
b) He doesn't have enough hands to make a reliable sample set (< 5000)
c) He's only remembering hands with flushes that beat him

Can you help me figure out the odds of:

- a flop having 2 (or 3) of the same suit (not related to holding two suited cards and flopping a 4-flush)
- having 3 or 4 to the flush on the turn
- A complete board having 3, 4, or 5 of the same suit.

After that, I'd like to try and figure out something that would show how reasonable it is to see X "flushes" in Y hands, and what the standard deviation (is that right?) would be.

I'd like to be able to show him that he's full of baloney on this with some hard math, if I can.

Thanks in advance.

[steveyz]

This is all assuming you don't know what anyone is holding:

Odds of a suited flop (all 3 of 1 suit):
52/52 * 12/51 * 11/50 = 5.18% (or 18.3:1 against)

Odds of 2 of a suit on a flop:
3 * 52/52 * 12/51 * 39/50 = 55.06%

Odds of a rainbow flop (all 3 of different suits):
52/52 * 39/51 * 26/50 = 39.76%

Having 4 to the flush on the turn:
52/52 * 12/51 * 11/50 * 10/49 = 1.056%

Having 3 to the flush on the turn:
4 * 52/52 * 12/51 * 11/50 * 39/49 = 16.48%

Complete board having 5 of same suit = 0.198%
4 of same suit = 4.29%
3 of same suit = 32.62%

---------------------------

As you can see, the odds having a 2 to a suit on the flop is greater than the odds of a rainbow flop, something that is probably not intuitive for most people. Also note that almost 1/3 of complete boards will have a flush possibility.

[MrDannimal]

Thanks man.

[uuDevil]

OK, I'll take a shot:

Note that there are several ways to do these kinds of problems.

In the following, assume you don't look at your hole cards (or aren't sitting in the game).

P(flop 3 of one suit)= (13/52)(12/51)(11/50)*(4 ways to pick a suit)=.051765, or 18.36:1 against.

P(Flopping 2 of 1 suit)= 1-(P(rainbow flop)+P(3 suited flop))=1-((52/52)(39/51)(26/50)+.051765))= .5506, or 0.82:1 against.

P(turning 3rd suited card)=P(2 suited flop)*(11/49)= 0.120, or 7.33:1 against.

P(turning 4th suited card)=P(3 suited flop)*(10/49)=.01056 or 93.66 against.

P(5 suited board)= P(turning 4th suited card)*(9/48)=.001981 or 503.85:1 against.

Now, I'm just going to consider POSSIBLE flushes, not actual made flushes, because the number of actual flushes will depend on how players play, not just how the cards come out.

The number of possible flushes in y hands should have a binomial distribution, Where the p=probability of a success (possible fl on bd), n= number of trials (hands), x= number of successes (possible fl on bd), and the standard deviation is given by sqrt(np(1-p)).

The binomial distribution is defined by
b(x;n,p)=C(n,x)*(p^x)*(1-p)^(n-x),
where C(n,x)= n!/((n-x)!x!)

The probability of the board being 3, 4 or 5 suited is 0.371 (1.7:1).

A spreadsheet program like MS Excel is convenient for doing these kinds of calculations. So in EXCEL, P(x possible flushes in n hands)= BINOMDIST(x,n,.371,false).

For example, P(x=3 possible flushes in n= 10 hands)=
BINOMDIST(3,10,.371,false)= .239 (3.2:1)

In 5000 hands, you would expect there will be 1855 possible flushes, with a SD= sqrt(5000*.371*(1-.371)=34. This means that if you took many 5000 hand samples, about 2/3 of the time you would find between 1821 and 1889 possible flushes.

Note that by these calculations, you would expect a possible flush more than once every 3rd hand!

But flushes will not actually occur this often because many hands that would have made flushes by the river will have been folded before then. I'll leave it to you to consider those possibilities....

Note however, that it is possible your friend may be noticing a real effect, since the number of flushes he sees will depend on how players behave. If several players will play any two suited or will stick around to catch a backdoor flush, then he will see many flushes.

I could easily have made an error somewhere. If so, hopefully someone will correct me. But this should get you started.

WHEW!! /images/graemlins/smile.gif