Flush draws

[pokeherguy]

Since all this stuff with Fa La La La La.... La la la la about the deal and the return card I've got something I'd like to know. I know I'm in the right place because I'm yet to see a math question involving video poker go unanswered here. How often should a player see 4 cards to the flush off the deal ?

[cddenver]

New2vp posted some "4 to" numbers recently in the WM - RS meeting thread.
 
4 to Flush:  85,512 possible dealt combinations, or roughly once every 30 hands.  That number doesn't include 4 to SF or 4 to RF combinations.

[pokeherguy]

I guess I missed that, thanks for the info. The 4 card to the flush off the deal has been one of those hands that caught my attention a while back when I started playing at a new casino. After playing for many years at a Tunica casino I can truly say the only thing that stood out was how rare the royal flush showed up (which is rare anyway). With the new casino the 4 card flush draw kind of snuck up on me. It was one of those deals where I was seeing it but not really paying attention at first. The more I played it became more noticeable and got to where I could almost predict how it would play out. With the info that you provided (1 every 30) and doing some quick math and based on 800 hands per hour I come up with around 26- 4 card flush draws per hour in which I should convert about 5. Unless it was all just a bad dream I got far more than that 26 per hour figure. Maybe its time I did my own little experiment.

[cddenver]

Unless it was all just a bad dream I got far more than that 26 per hour figure. Maybe its time I did my own little experiment.
 
Just like anything else that has to do with VP, the average value is good, but if you collect info during your play you'll see your actual occurrences coming in "runs" of different sorts.  I put together a little spreadsheet that shows what your runs for dealt 4 to Flushes should look like.  The point of this is to show that if you're looking for information from your actual play to match an average value you may well see funny numbers in the short term:
 
 10% occur within  4 hands of previous dealt 4 to Flush.
 20%               7 hands
 30%              11 hands
 40%              16 hands
 50%              22 hands
 60%              28 hands
 70%              37 hands
 80%              49 hands
 90%              70 hands
100%             436 hands
                   
This sheet was set up to actually deal hands so these numbers aren't "pure math" (perfect) ones, though these should be close to theoretical values.  I set up the sheet to run until I had 10,000 dealt 4 to Flushes.  Total hands dealt were 308,860.  So the average interval between dealt 4 to Flushes for this sample was 30.89 hands.  63.32% occurred in less than the average interval, if I round it to exactly 31 hands.
 
I like that maximum "dry spell" of 436 hands between dealt 4 to Flushes.  That's about half an hour of single line play!    If I ran that spreadsheet again that high value would change, sometimes ending up higher, and sometimes lower.  That's the problem with doing brute-force tests like this instead of doing it with pure math.  In this test my next-to-highest value was a much more reasonable 280 hands. 
 
That reminds me.  When New2vp put out his expected ranges the other day for what WM should see for flips in his upcoming test, assuming an 800 hand session, I decided to put a sheet together to play that scenario out using the parameters of the test, so I guess I'm still in a simulation mood.  I actually simulated about 50 sessions for that, and in every simulated session the number of "flip" opportunities and then actual "flips" fell within those expected ranges.   

[New2vp]

[cddenver]

Pure math will corroborate cd’s simulation as very realistic and accurate for the numbers up to and including 90%. If we see the probability of a 4-flush as about 0.0329 (85512/2598960), and we wish to determine the graph associated with a random variable equal to the number of draws until the next 4-flush, we can calculate this theoretically fairly simply.

 
Golly!    Nice to see life imitating art, again.  When you posted your reply I realized I had "cheated" in my simulation.  After I had my sample of 10,000, I sorted them by hand interval and took the easy path of deciding that the interval in the 1,000th spot was my 10% cutoff, the interval in the 2,000th spot was my 20% cutoff, etc.  That's why my numbers are integers while the true results are going to be fractional intervals.  I was also a bit sloppy on that top end; instead of listing a 100% value I should have cut it off at 99% (138 hands in my sample, which matches your number) or 99.9% (213 hands in my sample, close to your value of 207).
 
You used a pure math method to get your numbers without ever touching a deck of cards or playing a single hand; if I fixed my sim and reran with a large enough sample that would be another method of getting the numbers without walking into a casino.  That's an approach I'm more comfortable with, as I don't trust my ability to do the math correctly.  Another method would be half-and-half, using a smaller sample of actual results and then extrapolating or extending that to the longer term.  That's what will be done for the upcoming WM-RS meeting.  And they're all valid methods, when done correctly.   There's another method that's much more work, and also unnecessary.  That would be sitting in a casino and playing out 10K's or 100K's of hands testing for different things. 
 
If we did a "thought experiment" and assumed a world where casinos not only tracked our total play but all individual hands played and allowed us to get that info at the end of the year for a personal database, THAT information could be used to get the same results as well.
 
I don't track information in my real play down to this level, but I do for intervals between quads/SF's/RF's.  I've got graphs in the form you mentioned, where I have curves showing both my actual intervals between nice hits and those generated in sims, and they overlay each other at almost every point.  So I'm a believer in the randomness of the machines, and also in the effectiveness/appropriateness of taking a pure math approach (or simulation results) and applying them to real play. 
 
 

[pokeherguy]

Thanks for the help guys next time I need some help with my 10 year olds math I know where to turn. Now comes the hard one Rainman, how much does a house cost?

[Eduardo]

Thanks for the help guys next time I need some help with my 10 year olds math I know where to turn. Now comes the hard one Rainman, how much does a house cost?

Wow, ask a question then rip people when they answer it? That's cold, yo.

[New2vp]

Thanks for the help guys next time I need some help with my 10 year olds math I know where to turn. Now comes the hard one Rainman, how much does a house cost?Sorry, I should have indicated that my target audience was not people that needed help with their 10-year-old's math.  Housing costs are dependent on supply and demand, location, available schooling, size of lot and rooms, and a whole lot more factors than I could possibly simplify to a fifth-grade audience in the confines of a single post.I AM teasing as I presume you were in your post.  I know I get carried away but I do try and include enough information so that those inclined to read the details can understand what I've tried to convey.  For others, I simply figured they'd skip the hard parts like many do in school.No blood, no foul!

[cddenver]

Wow, ask a question then rip people when they answer it? That's cold, yo.
 
He must have been referring to New2vp.    My approach involved very little math, which is why I did it that way.  My smart pills are on backorder.