What are the chances of royal flushes in various games?

Question: Video poker and at least two table games have jackpots for royal flushes. What are the odds against pulling such a hand?

Answer: At the tables, jackpots are paid for royal flushes in Caribbean Stud and Let It Ride. In both cases, players get five cards with no discards or draws. The odds are therefore identical in the two games and are 649,739-to-1. This is figured by determining that 52 cards can be combined into 5-card hands in 2,598,960 different ways. Of these, only four are royal flushes. The probability is therefore 4/2,598,960, which equals 1/649,740. This means one way to win out of 649,740 total hands. Expressed as odds, it's 649,739 losses to every win.

The chances are different at video poker for several reasons. For instance, after you see your five initial cards, you can hold some and replace the rest; this means you can form the royal flush from as few as five cards to as many as 10. There's also an element of choice in that you could break a low pair and hold a single picture, sacrificing "expected win" but leaving open the chance of a royal. And, of course, the odds depend on details such as wild cards. Here are some typical values, based on playing strategies which maximize expected win.

jacks or better, nothing wild.....40,160-to-1
deuces wild (natural royal).......46,728-to-1
deuces wild (royal with deuces)...525-to-1
joker wild (natural royal)........39,619-to-1
joker wild (royal with joker).....10,309-to-1