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Ask the Slot Expert: Split Card calculations: Straight Flush - Part 23 July 2024
Let's finishing calculating the number of ways to make a Straight Flush in Split Card Deuces Wild when we're dealt Q♦ 2♠/2♠ 2♥ T♦ 8♥ and we hold the deuces. The easiest way to do the calculations is to look at each straight sequence separately. We'll look at combinations we can be dealt with the lowest card in the sequence and find the number of ways we can fill the fifth position without getting a higher-paying hand or double-counting a combo. We get a little bit of a break. We only have to do sequences beginning with A, 3, 4, 5, 6, 7, 8, and 9. We did the sequence 34567 last week. It's the easiest one to do because it's not affected by the cards we discarded so the numbers are the same for all four suits. For completeness, I included those results in the table below. The Tot Ways column shows the total number of ways to get a Straight Flush in the sequence specified. The Ways column shows the number of ways to get a Straight Flush in the suits specified. The Suit(s) column shows the suit or suits for which the details that follow apply. The Deal columns show the two or three cards (suited) we were dealt. The next set of columns shows the number of cards of each rank we have to remove from the cards remaining in the deck to avoid a higher-paying hand or double-counting a combination. The next-to-last column is "Fill", which gives the number of ways to fill the fifth position without getting a higher-paying hand or double-counting. For the two-card combos, the column is 43 minus the number of cards we removed. (Remember why it's 43? There are 47 cards left in the deck, minus two deuces that would give us a higher-paying hand [4 Deuces], minus the two cards dealt.) For the three-card combos, it's 1. There's only one way to get dealt those three cards. The final column is Tot. It is the sum of the numbers in the Fill column. Let's look at A2345. We have to remove the other aces from the deck because we don't want 5 of a Kind. Likewise, we have to remove the other threes for A3, the other fours for A4, and the other fives for A5. For A3, we have to remove the suited four and five, because we're handling A34 and A35 later. For A4, we remove the suited three and five and for A5, we remove the suited three and four. We also have to remove the suited ten, jack, queen, and king to prevent a Wild Royal. There are 31 ways each to make a Straight Flush with A3, A4, or A5, plus one way each for A34, A35, and A45 in clubs, hearts, or spades. That gives us a total of 96 for each suit. These results apply to three suits, 3(96) = 288. Diamonds is almost the same, but we have already discarded the ten and queen. That gives us a total of 102 ways for diamonds. The total for all suits is 390. The other sequences follow the same procedure: Remove the cards of the same rank we were dealt because that would give us 5 of a Kind. Remove the other suited cards in the sequence left in the deck because we'll take care of combinations with them in other entries. Remove any suited cards that can give us a Wild Royal. Avoiding a Wild Royal also eliminates some of the three-card combos we have to look at in 789TJ, 89TJQ, and 9TJQK (e.g., 7TJ, 8JQ, 9QK). Do a suit separately when we have already discarded one or more of the cards we need to remove.
We'll handle 4 of a Kind next week. 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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