Three to a Royal Flush

 

Is it correct to draw for a Royal Flush when we have 60% or 3/5 cards?

Let’s look at some examples and see what our math says.

ASSUMPTION 1: We’re not using real money. This is for entertainment only. Use your own math for real money gambling decisions.

ASSUMPTION 2: We’re playing Jacks or Better using the Cool Cat MAX COINS payout schedule:
Royal Flush 4000
Straight Flush 250
Four of a Kind 125
Full House 30
Flush 25
Straight 20
Three of a Kind 15
Two pair 10
Jacks or Better 5

EXAMPLE I: We’re dealt Qc Jc Tc Jd 2s

Option 1: We keep the Jacks and discard the Qc, Tc and 2s. Here are the expected value numbers:

Poker Hand Combinations Possibilities % Payout EV
Jacks or Better 11,559 16,215 71.30% 5 3.56
Two Pair 2,592 16,215 15.99% 10 1.60
Three of a kind 1,854 16,215 11.43% 15 1.72
Full House 165 16,215 1.00% 30 0.31
Four of a kind 45 16,215 0.28% 125 0.35
TOTAL 16,215 7.54

 

On the other hand, we could hold onto the Qc, Jc, and Tc and go for a Royal Flush among other possibilities. What does the math say?

Poker Hand Combinations Possibilities % Payout EV
Non-Qualifying Hand 756 1,081 70.00% 0 0
Jacks or Better 207 1,081 19.15% 5 0.96
Two Pair 21 1,081 1.94% 10 0.19
Three of a kind 7 1,081 0.60% 15 0.10
Straight 45 1,081 4.20% 20 0.83
Flush 42 1,081 3.90% 25 0.97
Straight Flush 2 1,081 0.20% 250 0.46
Royal Flush 1 1,081 0.09% 4,000 3.70
TOTAL 1,081 7.21

 

As you can see, our math says to stay with the pair of Jacks and not go for the Royal Flush even though we’re 60% of the way there.

EXAMPLE II: We’re dealt Kc Qc Jc Jd 2s

Option 1: We hold the jacks and discard the Kc, Qc, and 2s. Here are the expected value numbers:

Poker Hand Combinations Possibilities % Payout EV
Jacks or Better 11,559 16,215 71.30% 5 3.56
Two Pair 2,592 16,215 15.99% 10 1.60
Three of a kind 1,854 16,215 11.43% 15 1.72
Full House 165 16,215 1.00% 30 0.31
Four of a kind 45 16,215 0.28% 125 0.35
TOTAL 16,215 7.54

As you can see, these numbers are identical to option 1 of example I.

Option 2: We hold the Kc, Qc, and Jc. Here are the expected value calculations:

Poker Hand Combinations Possibilities % Payout EV
Non-Qualifying Hand 660 1,081 61.05% 0 0
Jacks or Better 318 1,081 29.42% 5 1.47
Two Pair 21 1,081 1.94% 10 0.19
Three of a Kind 7 1,081 0.60% 15 0.10
Straight 30 1,081 2.78% 20 0.56
Flush 43 1,081 3.97% 25 0.99
Straight Flush 1 1,081 0.09% 250 0.23
Royal Flush 1 1,081 0.09% 4,000 3.70
TOTAL 1,081 7.24

Again, the math tells us that we should hold the two jacks and take the guaranteed win.  However, as with the first example, the difference in expected values of our two options is very low.