I was at Maryland Live! casino near Baltimore recently, and they have some interesting bonuses and jackpots. During this trip, they were running a special “High Hand” jackpot, where the highest hand anywhere in the poker room each hour gets a payout of $2,500. In addition, they always have a Royal Flush Bonus, where anytime a player gets a royal flush using both hole cards, that player gets a $500 bonus and each other player at his or her table gets $100. My friend Brian once got the table share.

So there I am, playing $2/5 no limit with about $600 on the table, and I call a pre-flop raise to $20 from the cutoff seat with Jc Tc. The button also calls.

Flop ($60): Ac Kc 4h

A royal flush draw. But I’ll only get the bonus money for the Royal Flush and the High Hand if it actually hits. What is the optimal way to play this?

The pre-flop opener checks, I check, and the button bets $30. Opener folds. I call. I have a 2nd nut flush draw, gutshot straight draw, and one-outer to a Royal Flush. So the odds look like this:

Pot: $90

Amount to call: $30

Let’s assume I’m behind here, and my opponent has a hand like Ax or Kx.

I’m getting 3-to-1. I’ll hit the Royal Flush 1-in-47 times, and win $2,500 + $500 + $90 (assuming I get no further action).

I’ll hit a lesser flush 8-in-47 times, and a straight another 3-in-47 times (cannot count Qc twice), and win $90.

I’ll lose $30 the remaining 35-in-47 times, but may get another chance at the river card.

Here is the math:

1/47 x $3,090 = $65.75

11/47 x $90 = $21.05

35/47 x ($30) = ($22.35)

Add these up for Expected Value of $64.45. That’s positive EV, so calling is correct.

Without the Royal Flush and High Hand bonuses, the Qc result is the same as any other club, so the math is:

12/47 x $90 = $23.00

35/47 x ($30) – ($22.35)

Net EV is $0.65. Just a borderline call.

The turn card doesn’t help me, nor does it pair the board which might give the villain a full house, so the odds change only slightly.

Now I check again and he bets $45, into a pot that is now $120. Since another card has been revealed, the denominator is now 46 instead of 47. (We start with 52 cards. Subtract my 2 and the 4 community cards. The river will be one of the 46 remaining unknown cards. Yes, it is possible that the card I want is already in the muck pile, but those cards are all part of the unknown 46.) It will cost me $45 to try to win a pot that is now $165 including the villain’s turn bet, plus the Royal Flush and High Hand jackpots if the Qc hits.

1/46 x $3,165 = $68.80

11/46 x $165 = $39.45

35/46 x ($45) = ($34.25)

Add these up and the EV is $74.00. Proper to call.

Without the jackpot money, it looks like this:

12/46 x $165 = $43.05

35/46 x ($45) = ($34.25)

Net EV is still positive at $8.80, so calling is still a correct play.

Lastly, we need to consider the impact if I were to bet, or check-raise on the flop or turn. If I have any fold equity (value that I gain by winning the hand when the villain folds), how does that change the overall EV?

First of all, it must be observed that if I become the aggressor and get the villain to fold, I cannot win the Royal Flush or High Hand bonuses, a combined $3,000, **WHICH I REALLY, REALLY WANT TO WIN **(really, I do), as I won’t get to see another card. But let’s do the math anyway.

Part of the challenge is that we don’t know how often he will fold. We’ll look at 4 scenarios:

Scenario 1 – I check-raise the flop and he folds 1/3 of the time.

Scenario 2 – I check-raise the flop and he folds 2/3 of the time.

Scenario 3 – I call the flop, then check-raise the turn and he folds 1/3 of the time.

Scenario 4 – I call the flop, then check-raise the turn and he folds 2/3 of the time.

In each case, we’ll further assume that my check-raise bet size is 4x his bet on that street.

Ready for some math?

**Scenario 1:**

One-third of the time, he folds and I win $90.

The remaining two-thirds of the time, he calls my raise to $120. I can win the $60 that was in the pot pre-flop plus $120 more, so…

1/3 x $90 = $30

2/3 x 1/47 x $3,180 = $45.10

2/3 x 11/47 x $180 = $28.10

2/3 x 35/47 x ($120) = ($59.55)

Total EV is $43.65. This EV is lower than my calculation for just calling on the flop (which was $64.45), so calling is the better option – heavily influenced by the jackpots. Take away the jackpots, and now:

1/3 x $90 = $30

2/3 x 12/47 x $180 = $30.65

2/3 x 35/47 x ($120) = ($59.55)

Total EV is $1.10. Paltry, but better than the $0.65 EV of calling and no jackpots.

**Scenario 2:**

Now he is folding 2/3 of the time to my check-raise to $120 (i.e., 4x his bet of $30):

2/3 x $90 = $60

1/3 x 1/47 x $3,180 = $22.55

1/3 x 11/47 x $180 = $14.05

1/3 x 35/47 x ($120) = ($29.80)

Net EV is now $66.80. Whoa Nelly! Even the the jackpots that I forego when I can make him fold, the EV is now higher than the EV of calling and chasing the jackpots. If (and it is a big IF) this villain would really fold as much as 2/3 of the time to a 4x check-raise, that becomes a better play than calling and chasing. Of course, we don’t know what he has, nor have we played with this particular villain for very long.

**Scenario 3:**

I call the flop. Now the pot is $120, I check, he bets $45 and I check-raise 4x to $180. One-third of the time, he folds and I win $165.

The remaining two-thirds of the time, he calls my raise to $180. If I hit one of my outs on the river, I can win the $120 that was in the pot after the flop betting plus $180 more, so…

1/3 x $165 = $55.00

2/3 x 1/46 x $3,300 = $47.85

2/3 x 11/46 x $300 = $47.85

2/3 x 35/46 x ($180) = ($91.30)

Total EV is $59.40. This EV is also lower than my calculation for just calling on the turn (which was $74.00), so calling is still the better option due to the huge jackpots. Take away the jackpots, and now:

1/3 x $165 = $55.00

2/3 x 12/46 x $300 = $52.15

2/3 x 35/46 x ($180) = ($91.30)

Total EV is $15.85. Again this is slightly better than the $8.80 EV of calling on the turn with no jackpots.

**Scenario 4:**

Now he is folding 2/3 of the time to my turn check-raise to $180 (i.e., 4x his bet of $45):

2/3 x $165 = $110.00

1/3 x 1/46 x $3,300 = $23.90

1/3 x 11/46 x $300 = $23.90

1/3 x 35/46 x ($180) = ($45.65)

Net EV is now $112.15. Once again, with the greater fold equity, this becomes a better play than chasing the jackpots.

So what finally happened? Of course, the river was a total brick and we both checked. He had AQ, including the Qc which was my gin card, and wins the pot.

Given what we now know about his hand, how often would a typical, regular casino $2/5 no limit Holdem player fold to a flop or turn check-raise? Obviously I’m putting a lot more chips in there, and we now know the only way I could have won that pot was to make him fold. But would he fold? If we repeated this hand 100 times, would he fold often enough for my aggression to be profitable?