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# Ask the Slot Expert: More likely to get dealt Royal on throwaway hand?

17 July 2024

Whenever I throw away a hand at NSU, I remind myself that the bright side is that I'm sometimes more likely to get a Royal now.

I've been thinking that for a while and I decided that it's about time that I did the math to see whether it is true or just wishful thinking.

The chances of being dealt a Royal are the same for any video poker game that uses a 52-card deck: There are 2,598,960 5-card hands that be dealt from a 52-card deck; 4 of those hands are Royals; your chances are 1 in 649,740 (2,598,960/4) to be dealt a Royal.

Now let's look at our chances after we've thrown away five cards. There are 1,533,939 5-card hands that cen be dealt from a 47-card deck. How many of those are Royals?

It depends on what we discarded. In the worst-case scenario, we might have discarded three royal cards, say, T♠Q♦K♥. That leaves us with one suit in which we could get a Royal, so our chances are 1 in 1,533,939. That's definitely less likely than being dealt a Royal.

How about if we discard two royal cards, e.g., T♠Q♦? Now we can make a Royal in two suits, so our chances are 1 in 766,969.5 (1,533,939/2). We're still more likely to be dealt a Royal.

Now let's look at discarding only one royal card. We can make a Royal in three suits, so our chances rise to 1 in 511,313 (1,533,939/3). Now we've crossed the line and we're more likely to make a Royal than to have been dealt one.

If we don't discard any royal cards, our chances are 1 in 383,484.75 (1,533,393/4).

The following chart summarizes these numbers:

Dealt
1 in 649,740
1 in 1,533,9391 in 766,969.51 in 511,3131 in 383,484.75

A few days ago, a friend was playing 9/6 Jacks and mentioned that he was getting a lot of throwaway hands. I said that at least he was more likely to get a Royal now than to be dealt a Royal.

Was I right?

The numbers in the chart above still apply. The chances are 1 in 649,740 to be dealt a Royal.

The Discard 3 scenario doesn't apply. You would never throw away three Tens, Jacks, Queens, Kings, or Aces. You also wouldn't throw away a pair of those ranks. If you don't have Three of a Kind or a Pair, then at least two of those cards must be high cards, and you would never throw away all of the cards in that hand. There's no throwaway hand in which you discard three royal cards.

Likewise, Discard 2 also does not apply. You wouldn't throw away a pair. In an unsuited sequence, at least one of the cards must be a high card. It too is not a throwaway hand.

The Discard 1 scenario sometimes applies. You would never throw away all of the cards in a hand with a Jack, Queen, King, or Ace, but you might toss them all with a hand that had a 10. We might discard a card that makes it impossible to get a Royal in one suit.

If all of the cards we discard are 2 through 9 (Discard 0) or we discard one 10 (Discard 1), we are still more likely to be dealt a Royal from the cards remaining in the deck than to have been dealt a Royal in the beginning of the hand.

I went online to verify my methodology and results about "throwaway Royals." I found some confusion because posters were answering different questions.

One poster said that you were much less likely to be dealt a throwaway hand and then draw a Royal than to have been dealt a Royal in the first place. Note that his scenario starts at the Deal. His calculation is P(throwaway hand) x P(draw Royal).

That's not my scenario. For me, P(throwaway hand) = 1 because it has already happened. I'm just looking at getting a Royal on the Draw.

Frankly, I don't care how I get a Royal. I don't care if it is dealt or if I draw one, two, three, four, or five cards and get one. I would just like to get another one. Soon. It's been a while.

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