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Ask the Slot Expert: Split Card calculations: 5 of a Kind

19 June 2024

By John Robison

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

So far, I've calculated the EV of option 1. It was easy because the only hand I could have achieved that pays better than a Wild Royal is 4 Deuces. Of the 47 cards remaining in the deck, two of them are the deuces that improve my hand and the other 45 leave me with the Wild Royal, which I was quite happy to have.

Hold: Q♦ 2♠/2♠ 2♥ T♦
HandPaysWaysTotal Pays
4 Deuces100022000
Wild Royal125455625
Total7625
EV=Total/47162.23

Analyzing the second option turned out to be much harder than I anticipated, mainly due to having wild cards. After some missteps, I've gotten this far:

Hold: 2♠/2♠ 2♥
HandPaysWaysTotal Pays
5 Deuces20004590,000
4 Deuces100019801,980,000
Wild Royal1251357169,625
5 of a Kind80
Straight Flush50
4 of a Kind20
Total
EV=Total/16,215

This column has the calculations for the first option and for 5 Deuces and 4 Deuces. This column has the calculations for Wild Royal.

Let's try calculating 5 of a Kind this week. Unlike Einstein, who handled the special case of relativity a decade before the general case, we'll get our feet wet with the general case of achieving 5 of a Kind before tackling the special cases.

Our special cases are Q, 10 and 8 because we're discarding one card in each of those ranks. J, K, and A are also special cases because we have to be mindful of a draw that gives us the higher-paying Wild Royal.

We'll start with 9 (Noin?). We're already holding two cards that give us three deuces. We're drawing to fill three positions. To get 5 of a Kind, we need to draw two of the remaining four 9s left in the deck. We almost don't care about the card in the third position.

Let's start with looking at 9♣9♦. There are 47 cards remaining in the deck and we just drew two of them. There are 45 cards left to fill the third position.

But, we don't want to draw another deuce because that would give us 4 Deuces. And we really don't want one of the other two 9s in the deck because we want to avoid counting drawing three 9s multiple times. (It's sometimes easier to not over-count them in the first place than to figure out how many times they were over-counted.) So, we have 45-2-2=41 ways to fill the third position and have a 5 of a Kind in 9s with 9♣9♦.

The ways are the same regardless of the two suits drawn. We could draw 9♣9♦, 9♣9♥, 9♣9♠, 9♦9♥, 9♦9♠, 9♥9♠. That's the combination of 4 things chosen 2 at a time or 6.

We excluded choosing three 9s before to avoid counting them more than once. Let's take care of that now. The combination of 4 things chosen 3 at a time is 4. Another way to look at it is that we have four choices for which suit we will not draw in our trio of 9s.

So, we have 6 ways to choose two 9s times 41 ways to fill the third position plus 4 ways to choose three 9s, 6*41+4=250. There are 250 ways to get a 5 of a Kind in 9s.

The other ranks that are the same as 9 are 3, 4, 5, 6, and 7. That's seven ranks done (remember, we don't have to do 2). We're halfway there already.

Let's do 8 next. We could draw 8♣8♦ or 8♣8♠, or 8♦8♠. To fill the third position, we have the usual 43 cards (47 cards left in the deck less 2 we drew less 2 deuces) less 1 because we don't want the third 8 left in the deck giving us 42 ways. So, 3 different ways to draw two 8s times 42 ways to fill the third position plus 1 way to draw the remaining three 8s gives us a total 127 ways to get 5 of a Kind in 8s.

Now let's take care of the ranks that could lead to a Wild Royal. Fortunately, J, K, and A are the same as are 10 and Q.

Let's look at A♣A♥ first. From the 43 non-deuce cards remaining, we need to remove the cards that could complete a Wild Royal. Each suit has four cards (K, Q, J, and 10). We also need to eliminate the remaining two As in the deck. That gives us 43-4-4-2 or 33 ways for A♣A♥, A♣A♠ and A♥A♠.

Now let's look at the cases in which one of our aces is a diamond, e.g., A♣A♦. Here we have to deduct the four clubs that could give us a Wild Royal, the two diamonds that could give us a Wild Royal and the two remaining aces. That gives us 43-4-2-2 or 35 ways to get 5 of a Kind for A♣A♦, A♦A♥ and A♦A♠.

Summing up, three suit combinations give us 35 ways and three suit combinations give us 33 ways plus the 4 ways to get three aces gives us a total of 35+35+35+33+33+33+4=208 ways to get 5 of a Kind in aces. And kings and jacks, too.

Now let's do Q♣Q♥. We have to eliminate the four clubs and the four hearts that can give us a Wild Royal and the remaining queen, 43-4-4-1=34. There are 3 ways we can choose two of the three queens left in the deck times 34 plus 1 way to choose all three queens for a total of 3(34)+1=103 ways to 5 of a Kind in queens. And tens too.

Let's add 'em up. Adding up A+3+4+5+6+7+8+9+T+J+Q+K = 208+250+250+250+250+250+127+250+103+208+103+208 = 2457 ways to get 5 of a Kind.

Phew. Let's update our table.

Hold: 2♠/2♠ 2♥
HandPaysWaysTotal Pays
5 Deuces20004590,000
4 Deuces100019801,980,000
Wild Royal1251357169,625
5 of a Kind802457196,560
Straight Flush50
4 of a Kind20
Total
EV=Total/16,215

We'll do Straight Flush next week and finally have an answer as to which option is better.


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


 

Ask the Slot Expert: Son of still stumbling over Split Card calculations

12 June 2024
I ended last week's column with a challenge. I described a mistake I discovered in my formula for calculating the ways I could make a certain poker hand. Two of the terms were incorrect and I asked if you knew why. I gave the correct value, 43, as a hint. In my description, I wrote, "I hope those are the only mistakes." Guess what! Those weren't the only mistakes. ... (read more)
 

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♠ ... (read more)
 

Ask the Slot Expert: Figuring out Split Card video poker hand more difficult than I thought

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Last week I described a hand I had playing Split Card video poker. I wasn't sure if I had played the hand correctly and I said that I would eventually do the math to find the best play. Eventually was supposed to be this week. I thought I could do the math in an hour or so. I was wrong. Figuring out the expected value of the combination of cards you hold is usually fairly easy. ... (read more)

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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