Question of the Day - 22 May 2023

The answer to the original question is correct if there are no high cards (10 to ace) in the initial deal.  If high cards are discarded, then it is not possible to get a royal in that suit, so the odds will increase.

The additional odds provided for drawing 1 to 4 cards to the royal are correct because the royal can only be one suit.

As soon as you press the 'Deal' button the computer chooses ten (10) cards!
This leaves the deck with 42 cards.
You discard 5 cards and are immediately presented with 5 new cards.
When you discarded the 5 cards the deck has now grown to 47 cards.

IF 10 cards were not drawn at first then it would be possible for you to redeal the exact same 5 cards you discarded, that has never happened, not even one card of the 5 discarded because your second set of 5 cards were dealt at the beginning when you hit the 'Deal' button.

The initial deal of 10 cards has always been in place for the reason of not dealing back to you any of your discards.

Therefore, the true odds are not those quoted you because the odds should have been based upon the probability of a Royal being dealt within 10 cards out of 52.

In computer programming terms this was a better solution than having a trash can implemented to hold your discards and then hope the trash can releases all 5 cards back for the next deal.

I did once discard my first five cards and then received a Royal.
Unfortunately, I didn't have 'Max Bet' selected.

If you are in Washington State, the odds of getting a Royal after throwing away 5 junk cards is zero.

Conversely, the odds of getting a Royal after throwing away a Royal is 1.

Mufasa -

You're right. Normally, if you have nothing but a high card, you hold it. Five small cards, which include a ten would get discarded.

I believe the math is simply 3/4 of the five card number listed above: 1 in 287,613.


Jackie -

You're oversimplifying things. There are MANY times where you wouldn't discard all five cards, but you would have drawn a royal if you did.

The above calculations just give the odds of drawing the royal once you have x to the royal and decide to go for it. That would include going against basic strategy such as discarding a small pair, etc.


Kieth -

Washington state, as well as some other jurisdictions, have weird rules. Yes, the odds are 1 (I.E. 100%). 

YOU BET $383,484 $1.00 PER HAND, YOU HIT A ROYAL..
WHAT'S THE PAYOFF?  I'M READY TO BREAK THE HOUSE IF THE PAYOFF IS MORE THAN $383,484.

Jackie is both right and wrong. Older VP machines to use the "draw 10 cards" method. New machines deal 5 cards and continually shuffle the remaining 47 cards, when you hit "Draw," the machine deals the top 1-5 remaining cards in the shuffle based on how many are needed.

Either way, however, the odds are the same.

The actual answer really does depend on how the cards are determined. Are they, as Jackie says, 10 cards dealt? or are 5 cards dealt and then the random number generates 5 of the remaining 47? But the odds that the answer quotes are truly based on getting a royal dealt from the remaining 47 cards, which, as Mufasa pointed out, are different depending on whether or not your first 5 included a high card. (I would expect a lone 10, because any other single high card would be held, not discarded)

Ten cards at once vs five now and five later makes no difference. Some think the likelihood if having a certain card is based on the composition of the deck when the cards are dealt even after seeing other cards from that deck. It doesn't work that way.

Here's an example: Get a deck of cards and deal yourself one card face down. The probability of it being an ace is 1/13, or roughly 7.7% (4 aces per 52 cards). Now, turn over some of the undealt cards -- let's say 40 for this example. If the 40 cards contain four aces, the probably of the one dealt card being an ace is clearly no longer 7.7%. It is 0%, as there are no aces remaining. 

OTOH, if there were no aces in the 40 cards, there are 11 unknown cards in the deck and one unknown dealt card. Now the probability of the dealt card being an ace is 4/12, or 1/3...33.3%.

The same principle holds true for 1, 2, or 3 aces in the deck.

Ray - Your comment about a lone 10 being discarded is correct in Jacks or better games.  However, in dueces wild games, it is very likely that more than one high card is discarded, so 2, 3 or all 4 suits could be eliminated from possible royal results.

Assuming none of your tossed cards were 10 or above,  your odds increase (now 47 v 52) - but there is no further 'round' of drawing. 

It seems to happen quite often that you are dealt 3 cards to the royal and hold them, you get a 4th card on the draw but not the 5th.  That's quite the teaser, being one card away from hitting a jackpot.

Actually just experienced this last week at Lake Tahoe--tossed all 5 cards playing 10 hands at a time, and ended up getting the royal on one of them.  Wahoo!

My comment was intended for non-wild card games. Adding deuces to the mix changes the dynamic, but certainly aces or faces could be discarded in a deuces game.