Newsletter Signup
Stay informed with the
NEW Casino City Times newsletter!
Recent Articles
Best of John Robison

# Ask the Slot Expert: Still stumbling over Split Card calculations

5 June 2024

A few weeks ago I played Deuces Wild with the Split Card gimmick. Split Card sometimes gives you two cards in one position of your hand. Although it's possible to get 6-of-a-kind and other six-card hands, the only bonus hand is 5 deuces, which pays 2000.

I was dealt this hand: Q♦ 2♠/2♠ 2♥ T♦ 8♥.

I wasn't sure which combination was better to hold:

1. Q♦ 2♠/2♠ 2♥ T♦ and be guaranteed the Wild Royal and maybe get 4 Deuces
2. 2♠/2♠ 2♥ and take a chance on 5 Deuces

Analyzing the first option is easy. There is only hand I could achieve that would be better than a Wild royal -- 4 Deuces. There are 2 cards left in the deck that would give me the set of deuces. The other 45 cards left in the deck leave me with the Wild Royal.

 Hold: Q♦ 2♠/2♠ 2♥ T♦ Hand Pays Ways Total Pays 4 Deuces 1000 2 2000 Wild Royal 125 45 5625 Total 7625 EV=Total/47 162.23

So far, so good.

Let's move on to the second options, holding 2♠/2♠ 2♥.

The hands I can achieve are: 5 Deuces, 4 Deuces, 5-of-a-kind, Straight Flush, and 4-of-a-kind.

To get 5 Deuces, I need the 2 deuces left in the deck and 1 of the remaining 45 cards. There's only 1 way to get the 2 deuces and 45 ways to fill the final position. So there are 45 ways to get 5 Deuces and win 2000. That was easy.

Figuring out 4 Deuces is also easy. There are 2 ways we can choose 1 of the 2 deuces left in the deck and 990 ways to choose 2 of the other 45 cards, giving 1980 ways to get 4 Deuces.

The next hand in the paytable is the Wild Royal. Let's skip it for now. It's usually easier to figure out the formulas for set-based hands than for sequence-based hands.

To get 5-of-a-kind, we need to draw 3 cards of the same rank. There are 9 complete ranks (A,3,4,5,6,7,9,J,K), 3 ranks missing 1 card (8,T,Q), and deuces. So we have 9 choices of complete ranks and we have to choose 3 of the 4 cards left. We have 3 incomplete ranks and we have to choose all of those cards. There are 9 ways to choose a complete rank and 4 ways to choose 3 of the 4 cards in the rank and 3 ways to choose an incomplete rank and only 1 way to choose all of the cards remaining in that rank. So 9x4 + 3x1=39 ways to get 5-of-a-kind.

Well, not so fast. Do we really need to draw 3 cards in a rank to get 5-of-a-kind? Don't we already have 3 deuces?

I was about to show my work for counting the number of ways to get a wild royal when I realized I had a mistake there. And then I realized that my 5-of-a-kind calculation was wrong too. Only 39 ways to get 5-of-a-kind seemed a bit low.

We need to draw only 2 cards of the same rank to get 5-of-a-kind using the 3 deuces we already have. Let's try the calculation again.

There are 9 ways to choose a complete rank, and 6 ways to choose 2 of the 4 cards remaining in that rank and 45 ways to fill the remaining position.

There are 3 ways to choose an incomplete rank and 3 ways to choose 2 of the 3 cards remaining in that rank and 45 ways to fill the remaining position.

The formula is: 9*C(4,2)*45 + 3*C(3,2)*45.

That's still not right. I made two mistakes or one mistake twice (and I hope those are the only mistakes). The 45 is wrong. Do you know why? Here's a hint. It should be 43.

If you would like to see more non-smoking areas on slot floors in Las Vegas, please sign my petition on change.org.

John Robison

John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots
John Robison
John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots