Odds Of Drawing A Royal Flush Are 1 In 649,740

January 19, 2012 by staff 

Odds Of Drawing A Royal Flush Are 1 In 649,740, In Vegas, some bets are safer than others. Doubling down in blackjack, outside betting in roulette (putting money on several numbers instead of one), or just sticking to craps. These involve relatively low risks, since they anticipate some of gambling’s more common outcomes.

A royal flush is not one of the common ones. It’s one of gameplay’s quintessential rarities. Everyone is aware of it, but just how rare is it? The odds a person will be struck by lightning in a year are 1 in 835,500. Is it rarer than that?

Actually, it’s more likely.

What about the odds two baseball players on a team’s active roster share a birthday? The odds of that are 1 in 1.76, or 57%.

A royal flush is rarer than that.

These odds may seem inordinately high, much higher than the 25/365 (7%) you might expect, but they are simply less intuitive than royal flush odds, which anyone knows could not possibly be a simple 5/52 (9.6%).

Combinatorics is what’s at work here: Draw 4 dots on a piece of paper and connect every dot to every other dot, and you should see 6 lines on your paper. Do the same for 5 dots, and it’ll take 10 lines; 6 dots need 15, and so on. By the time you get to 25 dots, 300 lines are required to connect them all-25 birthdays combine for a total of 300 pairings. The pairings are what raise the odds.

And what are the odds of a royal flush on the first draw? 1 in 649,740. You’re almost exactly as likely to die from appendicitis in a year as you are to draw a royal flush. Here’s why.
In five card draw, a royal flush contains 5 specific cards drawn from the 52-card deck. When you draw your first card, it has a 5/52 chance of being one of the cards needed for a specific royal flush. Assuming you got one, the next card you draw now has a 4/51 chance of being a card in your royal flush. The next card’s chance of fitting your royal flush is 3/50, then 2/49, and finally 1/48. The odds overall of drawing a royal flush may be gotten by multiplying these fractions together, and then multiplying that product by 4, since 4 royal flushes are possible, one for each suite:

4(5/52)(4/51)(3/50)(2/49)(1/48) = 480/311,875,200 = 1/649,740

Or here’s a slightly different rationale. There is a finite number of possible hands in poker: 2,598,960 of them. Only 4 of those hands are royal flushes. 4 from 2,598,960 is 1 from 649,740.

In 2007, player Dag Mikkelsen drew a royal flush in the World Series of Poker. Note that Texas Hold ‘Em is slightly different than five card draw: a royal flush is not dealt all at once, but rather as a pair of cards which matches up with the shared “river” cards. At the same event was former champion Greg Raymer, known for his holographic sunglasses. While Raymer did not draw his own royal flush at the time, he and Mikkelsen do have something in common.

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2 Responses to “Odds Of Drawing A Royal Flush Are 1 In 649,740”

  1. Jeff on January 19th, 2012 6:12 pm

    I think the explanation is screwed up.

    On the first card, it does not matter whether you get a face card in clubs or spades ro whatever. So loang as toyour card is a 10-Ace fo any suit, you can still draw a royal flush. there are 20 cards in a deck that will fit a potential royal flush on eth frist draw (10-A of spades, 10-A of hearts, 10-A of diamonds and 10-A of clubs). Thus, the first fraction should be 20/52, not 5/52. Thereafter the fractions are correct.

  2. Nick the Greek on January 19th, 2012 10:19 pm

    they multiple by 4 at the end, you multiplied by 4 in the beginning, same thing