Question of the Day - 05 May 2023

These odds are different in actual "real" world games played with multiple other "people."

What are the chances of being dealt 4 to a royal, and then drawing the 5th card, not just drawing the 5th card out of the 47 cards left.

In my case,  the odds of drawing the 5th card are zero. ;)

That's the question he answers. If you're dealt four to a royal, discard one, and receive another that makes a royal, the odds are 1 in 47. That's because there were initially 52 cards in the deck and five were used, leaving 47.

When playing with other people the odds are calculated quite differently as there are many other variables such as the number of players and where you sit in relation to the deal order. 

The first assumption is that you are playing a 5 card game vs a game like texas holdem or 7 card stud. 

Assuming you have 4 cards of the Royal and there are 4 other players - There are 25 cards dealt - and 27 undealt. The probability that your required card remains undealt is the first variable you need to calculate. So of the 25 dealt - you know there are 20 cards that might be the one you need and there is 0 possibility that you have access to that card. More players worse the odds. So still assuming your card is undealt then its 1/27 (3%) chance of drawing the needed card. These odds are then further reduced depending on where you sit on the deal and how many cards are drawn by other players before it gets to you. 

A machine has no personality but inherently you have better odds on a draw. 

If it is Joker game and the joker is not included in the original draw - it should be 2 in 48.  
If the joker is in the original draw, the odds are 1 in 48.
With a Joker in the deck is actually 53 cards.

I hit one a number of years ago at Pala casino in SoCal. Paid the same if it wasn't one. I hae the picture up on my Facebook page. Dealt the AQJ and had to draw the K10 all in diamonds. Once in a life time. 

Deke should have used the word "after", rather than "of" -- i.e. "Your chances AFTER being dealt four to the royal ..."

As he wrote, and most of the commenters understand, there are 47 (or 48) cards left, so that's pretty straightforward.

Also, we are talking about Video Poker -- one player, one deck of cards, and not a table game, with more than one player drawing from that one deck.

As for being dealt four to the royal, there are plenty of places to look up the chances of that, e.g., Mike Shakelford's or Bob Dancer's sites.

(Geek warning:  it's something called combinatorial analysis; here, it's the number of ways 52 (or 53) things that are selected four at a time)

You can't say 1 in 47 since there is only one card that completes the royal.
None of the other remaining 46 cards can do that but technically you are seeking that one card out of the 47. Therefor the true odds are 46 to 1.

Probability is defined as winners in (or over) chances while odds are typically written as losers to winners where chances=winners+losers. For example flipping a fair coin has winners=losers so odds are 1 to 1 (1:1) while probability is 1 in 2 (or 1/2). Rolling a fair die hoping for a specific number there is 1 winner and 5 losers so the probability is 1 in 6 (or 1/6) and the odds are 5 to 1 (5:1).

In this case there is 1 winner and 46 losers so the odds are 46 to 1 (46:1) and the probability is 1 in 47 (or 1/47). They describe the same thing but from a different perspective.