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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. Even my hint was a mistake.

First, a recap of the problem.

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. I also did okay on calculating the number of ways to achieve 5 Deuces and 4 Deuces in the second option. See last week's column. Disregard the 5-of-a-kind discussion. Please.

The 5-of-a-kind calculation is more complicated than I ever would have imagined. We'll get to it next week. This week let's tackle calculating the number of ways to get a Wild Royal.

Just as a reminder, we're holding 3 deuces and we have 3 positions to fill in the hand (one of the positions we're holding is a split card with 2 deuces).

We have two cases: diamonds and the other suits. We have to calculate diamonds separately because we are discarding two of the cards that can make up a royal.

Let's tackle diamonds first because we have fewer cards to deal with. We still have three cards that could appear in a wild royal (A♦, K♦, and J♦) in the deck. Normally I would follow this logic: pick 2 of the 3 royal cards left and fill the last position with one of the 45 cards remaining to give the formula C(3,2)*45. (C(3,2) means to calculate the number of ways to choose 2 items from a group of 3 when order does not matter, or the combination of 3 things taken 2 at a time. or choose 2 from 3.) That's not right, so let's use a different approach.

Let's start by filling the first two discard positions with A♦K♦. There are 45 cards remaining in the deck that could fill the last position, but we don't want all of them. Two of the cards are deuces and a deuce would give us 4 Deuces. We don't want a deuce, so there are only 43 cards we can fill the last position with and have a wild royal.

Now let's look at A♦J♦. You might be tempted to say that there again are 43 cards to fill the last position, but we didn't exclude J♦ in the A♦K♦ calculation. Let's avoid double counting A♦K♦J♦ now and say there are 42 cards left to fill the last position.

Let's do the same for A♦K♦ and say that there are 42 ways to fill the last position. We'll take care of drawing A♦K♦J♦ at the end.

There are also 42 ways to fill our hand when we draw K♦J♦.

How many ways can we draw A♦K♦J♦? That's an easy one. One. Pun intended.

What do we have? The formula is 42+42+42+1 or 3*42+1. Hm. That also happens to be C(3,2)*C(42,1)+C(3,3) (choose 2 from 3 times choose 1 from 42 plus choose 3 from 3). I guess I could have used combinations after all.

There are 127 ways to get a Wild Royal in diamonds.

I think we're ready now to tackle the other three suits. They have their A, K, Q, J, and T still in the deck. How many ways can we complete AK? There are 47 cards left in the deck, less the 2 we drew, less the 2 deuces, less the other 3 cards that could be in a royal. That leaves us with 40 ways.

We can draw AK, AQ, AJ, AT, KQ, KJ, KT, QJ, QT, and JT. That's 10 ways or C(5,2) (choose 2 from 5).

Now we have to handle the case where all of our replacement cards are part of a Wild Royal. That's C(5,3) or 10.

So, we have C(5,2)*40+C(5,3)=10*40+10=410 ways to get a Wild Royal in either clubs, hearts or spades. The total number of ways is 127+3*410=1357.

Next week we'll look at 5-of-a-kind.


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