Three to a flush in 9/6 Jacks-or-Better

I've got an optimal play question on a particular hand I faced on the Wizard's applet. I was betting five credits on the default payschedule, so that the payoffs would be 5-10-15-20-30-45-125-250-4000.

Here's the hand:

As 2c Th 3s Ks

I kept the suited AK, but this was recorded as an error. Seems I should have kept the same-suited trey as well. Here is the error entry:

HAND	BEST PLAY	EV	PLAY MADE	EV	ERROR VALUE
As 2c Th 3s Ks	A--3K	2.1869	A---K	2.1341	0.0528

My first thought was that the Th was the culprit. Discarding it reduces the probability of drawing to a straight. But that probability was pretty slim to begin with, and now I'm not so sure. I can't find mention in the wizard's optimal JoB strategy of drawing at 3-to-a-flush, ever.

Of course that error value amounts to 1% of a wager, on a situation that probably doesn't happen very often. But, out of academic curiosity--and before I let that curiosity drive me through a full analysis--does anyone know what's going on here?

That's odd. Must be a glitch. The correct play is to hold the A-Ks. I've never gotten an error on that before.

I had asked the Wizard for a one-coin JorB strategy (which he graciously provided in an earlier thread), and the holding of the third flush card is a legitimate play in that strategy. In order:

...
4 to a Straight (JQKA)
2 to a Royal Flush (JQ, JK, QK) *
3 to a Flush with 2 high cards [this is the case we are talking about]
2 to a Royal Flush (JA, QA, KA)
....

* 3 to a Flush with 2 high cards beats Suited JK or QK if the cards include a Jack, Queen & King

I had thought this was just a special one-coin strategy.

Either this strategy also applies to full-pay 5 coin play, or the Java Script game is determining its strategy based upon the 250 RF payout, not the 800 RF payout.

Only the Wizard knows for sure...

Thanks for this great find. A royal payout discrepancy between 4000 and 1250 (because I'm betting 5) times the 1-in-16215 royal probability translates to a change in EV of 0.1696, which is bigger than the error value I got.

However, I don't think this is what's going on. I duplicated the program's analysis sheet (on a different hand, since sadly I didn't have the foresight to capture one for the hand in question) and found that the EVs given are valuing the royal at 4000 when I bet 5.

I tried to go one better and calculate the 2.1341 EV that the error report gave me but, working by hand, I can't seem to arrive at the same figure. My calculation yields 2.8403 instead:

A---K	nothing	JoB	2Pr	3oK	Str	Fl	FH	4oK	StrFl	Royal	TOT
DRAWS	10012	5022	711	281	48	120	18	2	0	1	16215
PAYRATE	0	1	2	3	4	6	9	25	50	800	-
EXPECTED PAYRATE	0.00000	0.30971	0.08770	0.05199	0.01184	0.04440	0.00999	0.00308	0.00000	0.04934	0.56805
EV(on betting 5)	0.0000	1.5486	0.4385	0.2599	0.0592	0.2220	0.0500	0.0154	0.0000	0.2467	2.8403

What I really want is to see the app's analysis sheet on As 2c Th 3s Ks, but it's going to be a long time before that exact hand comes round again.

zapakh, I get the same numbers as you except I get 47 straights and 119 flushes (and 10014 garbage). I believe you are counting the royal flush 3 times. However, the effect on expected value is only 0.0031, so that doesn't help much.

Edit: You were betting 4 coins. If you apply my correction, and also use 4-coin payouts (including 250*4 for a royal), you get an EV of 2.1341. Also, the 4-coin EV of holding AK3 is 2.1869.

Thank you!

I can't believe I did that, but I can't argue with the evidence.

Thank you for spotting my error. The wizard's optimal strategy survives intact and I can get a good night's sleep.