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Ask the Slot Expert: Son of still stumbling over Split Card calculations12 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:
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. Send your slot and video poker questions to John Robison, Slot Expert™, at slotexpert@slotexpert.com.
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