Ask the Slot Expert: Things get wild with wild cards

I noticed something unexpected when I did the math for last week's column about the EV changing when you redraw a hand that has suited A-K versus suited A-Q versus suited A-J in NSU. Here is the table from that column.

The EV of holding the 2-card royal doesn't change, but the EV of redrawing does. It decreases as the second card in our 2-card royal decreases mainly because we're eliminating more ways we can make a straight or straight flush. When we discarded the king, we got rid of some of the king-high straights. When we discarded the queen, we got rid of some of the queen-high straights and also some of the king-high straights.

For example, when we redraw the hand with the king, there are 1492 [Note: I accidentally typed 1490 last week] ways to make a straight flush. When the hand has the queen, there are 1457 ways.

I expected the number of straight flushes and straights to decrease comparing redrawing the hand with A-K versus A-Q. What I didn't expect, however was the number of ways to make some other hands to change. For example, the number of ways to make four-of-a-kind increases from 21,725 for redrawing A♣ K♣ 3♦ 7♥ 9♠ versus 21,729 for redrawing A♣ Q♣ 3♦ 7♥ 9♠. I expected the number of ways to make the sequence-based hands to change, but I didn't expect the number of ways to make the set-based hands to change.

If I remember correctly from when I did these calculations for similar hands for Jacks or Better, the number of quads didn't change. Why does it change for NSU?

The world is nice, simple, and orderly for games without wild cards. When you have wild cards, things get -- well -- wild.

Let's try to find where the extra four quads are. Because NSU has wild cards, we don't need four cards of a rank to make a quad. We can have four-of-a-kind with a hand that has three deuces. Let's try to count these hands by rank to find the number of hands that have three deuces and an ace for quad aces, three deuces and a 3 for quad 3s, and so on.

Already there's a problem. Is a hand with three deuces and unsuited A and 7 four aces or four 3s? It's both and we have to be careful not to double count them.

We have to take a different approach to find where the extra quads are. Let's forget about classifying hands as quad aces, quad 3s, and so on, and just count the number of hands evaluated as four-of-a-kind grouped by how many deuces are in the hand.

I didn't expect to hit paydirt on the first line. Why are there more quads left in the deck when we throw away the hand with the queen?

I looked at all of the quad hands with three deuces for each throwaway hand. I made two lists: the quads I got when discarding the A-K hand and the quads I got when discarding the A-Q hand. I then looked at the hands that were in one set and not in the other. For example, 2♣ 2♦ 2♠ 7♠ Q♣ is in the set of hands from throwing away the A-K hand and 2♣ 2♦ 2♠ 7♠ K♣ is in the A-Q set. I matched these two hands because their first four cards are the same and they differ only in the fifth card, swapping Q♣ for K♣, and vice versa, based on which card was still in the deck.

I then looked for a pattern that was in the A-Q set and not in the A-K set. The hand 2♣ 2♦ 2♠ 8♣ K♣ has no analog in the A-K set.

What happens when we substitute the queen for the king in this hand? We get 2♣ 2♦ 2♠ 8♣ Q♣ and that could be four eights or four queens or, even better, a straight flush.

We found one of the four extra quads we can get when we throw away the A-Q hand. There are four ways to choose the three deuces in the hand. Tada! There are the four extra hands that give us four more quads when redrawing the A-Q throwaway hand.

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