#### Friday, March 23rd, 2012...08:13

## Royal Flush in the Family

tl;dr Last month I was part of an event 6 times rarer than winning the MEGA Millions Lotto jackpot.

My grandfather played poker with a core group of friends (to my knowlegde he did not play in card rooms or casinos) and taught me the basics of the game. We played Five-card draw with penny antes. Those instances and the occasional occasional video poker machine summed up my poker experience until the recent Texas Hold’em craze that started with Chris Moneymaker‘s 2003 World Series of Poker victory and the explosion of online gambling. I learned the ropes of that poker variant and play with friends and in casinos when the opportunity arises.

A few years ago my grandmother asked me if I wanted any of my grandfather’s belongings (he had passed away several years earlier). I remembered a framed royal flush, signed and dated by witnesses (I’m not sure whether he was playing Five-card draw at the time, and if so how many of the royal flush cards he was dealt vs. exchanged).

On February 28th I was in South Lake Tahoe for a company ski trip and sat down in the Harvey’s poker room with a couple of co-workers (no-limit Texas Hold’em, $2 small blind, $3 big blind). After playing for about an hour, our table of ten started to lose people, eventually becoming a table of five and disbanding. My co-workers moved to a table with two empty seats and I to a table with one (maximum of ten players per table).

After a few unworthy hands the dealer tossed KQ♠ (king and queen of spades) as my starting hand. I sat in fifth position. One person called and one folded in front of me, I called, two people called after me and both blinds joined in. With six players and $18 in the pot, the dealer flopped A♠K♦J♠. The blinds checked, the player in front of me bet $10. I called, the two players behind me called and the blinds folded. Four players left, $53 in the pot (dealer took $4 for the casino whenever the pot reached $40). The turn was 3♣. The person in front of me checked, I checked, the next person checked and then the last player bet $15. The player in front of me folded, I called and the player after me called. Three players left, $97 in the pot. The river was T♠. Holy cow, not only did I have a flush, but I had a royal flush, the highest possible hand in all of poker.

First to act and trying to keep my cool, I took a moment to consider my options. I could check, hoping one of the other two players would bet and then raise them. I could make a small bet, say $20-$30 and perhaps someone would call or re-raise me. Or I could make a sizable bet and it’s possible someone might think I was trying to buy the pot. The people remaining each had over $500 worth of chips in front of them and I had a little less than $200. I put out $75, about 75% of the pot and 40% of my stack in the hopes that someone else had hit something and thought I was attempting to buy the pot. The next guy thought for a while, but eventually folded and the other player quickly folded, oh well. I didn’t have to show my cards, but how could I not show a royal flush?

So I flipped over my cards and the table erupted. It also turned out the casino had a bonus jackpot which grew slowly based on the number of hands dealt and I received 5% of it for showing a royal flush. It wasn’t a boatload of money i.e. I learned the casino isn’t required to report payouts of under $600 to the government, but it was a pleasant surprise. I tipped the dealer about 10% of my bonus and then thought of my grandfather’s royal flush and asked if I could keep the deck of cards. The dealer quickly said that wasn’t allowed, gathered the cards and placed them in the shuffler as if nothing special had happened (he mentioned he had dealt a royal flush earlier in the day). I wanted my tip back.

I received some crappy cards for the next hand and immediately mucked them. As I finished stacking my winnings and lamented not having an item of memorabilia to go with my grandfather’s, not even a photo, something strange happend. The table erupted again as another player had just turned over a royal flush in hearts. The board had AKJT♥ (not sure about the fifth card) on it and they had Q♥ in their hand. This was a first for everyone on the table, including the dealer, two royal flushes in a row. People jokingly asked if the dealer was worried about getting fired, after all he had caused the casino to payout ~15% of the bonus today (he answered no).

My co-workers and I started wondering how often one could expect two royal flushes in a row during a game of Texas Hold’em. In the simplest scenario i.e. choosing 5 random cards from a standard 52 card deck, the odds of getting a royal flush are 1 in 649,350 hands. That number is generated via a branch of mathematics called Combinatorics. First one calculates the number of ways to select 5 cards from 52 (without replacing any during the selection process). There are 52 different ways 1 card can be picked from a deck, one for each card. That leaves 51 cards from which one can select another card, and after that card has been picked, 50 cards remain etc. So one might think the number of ways to select 5 cards from a 52 card deck is 52*51*50*49*48 = 311,875,200.

That calculation would be correct if one was looking for all the permutations of 5 card poker hands i.e. the order was important e.g. selecting A2358♣ was considered different from picking 35A82♣. In poker, combinations not permutations matter. Our original computation can be written using factorial notation i.e. 52!/(52-5)! = 52!/47!. It’s easy to work out with smaller numbers e.g. select 3 cards from a deck of 5 = 5*4*3 = 60 and 5!/(5-3)! = 5!/2! = (5*4*3*2*1)/2*1 = 60. In the case of choosing 3 cards from a deck of 5, one must also divide by 3! since any permutation works. As an example, for 3 items A, B and C there are 6 combinations, ABC, ACB, BAC, BCA, CAB, CBA . This is because there are X different items which can occupy the first position, X-1 for the next, etc. which leads to X!.

Getting back to the 5 cards out of 52 example our the number of possible 5 card hands in poker is 52!/(5!*(52-47)!) or 2,598,560. This is often written as (52 choose 5) and the math becomes N!/(K!*(N-K)!) for any (N choose K). There are 4 possible royal flushes so the odds of getting dealt one given 5 cards are 4 in 2,598,560 or 1 in 649,350. Given 7 cards as in Texas Hold’em, getting a royal flush of 5 cards should happen more often. The number of possible 7 card hands is (52 choose 7) = 133,784,560. If someone has a 5 card ace high straight flush, it doesn’t matter what the other 2 cards are, so we can simply multiply the number of different royal flushes, 4, by the number of ways to choose 2 cards out of the remaining 47 (52-5 since the 5 cards that make up a royal flush are already out of the deck). (47 choose 2) = 1,081, 4*1,081 = 4,324 making the odds 4,324 in 133,784,568 or 1 in 30,940.

At my poker playing peak, probably sometime in 2004 or 2005, I played online a few hours per week, at a friends house a few hours once a month and at a casino for a few hours once a year, so 3*52+3*12+3 = 195 hours of poker/year. If I saw 20 hands/hour, that’d be 20*195 = 3900 hands/year and I so perhaps I’d expect to get one royal flush roughly every 8 years.

But I hadn’t even witnessed a royal flush before; when playing at a table with multiple people, there is more than one 7 card hand happening at a time (often there are 8-10 players). So how often could I expect to witness a royal flush? One of my co-workers came up with this line of thinking. Royal flushes are unique in that no matter how many people are playing, only one person can have a royal flush per hand i.e. the probabilities for each additional player at the table getting a royal flush are mostly disjoint (mostly because if the royal flush appears in the 5 community cards, everyone at the table has a royal flush so that event should only be counted once). We’ve already calculated the odds of getting a royal flush given 7 cards, so for each additional player at the table (assuming they stay in to the river) the probability of witnessing a royal flush goes up by the probability of a single person getting a royal flush. Let’s set some variables:

- RF5 = p(royal flush|5 cards) = probability of a royal flush given 5 cards = 4*(52 choose 5) = 0.00000154
- RF7 = p(royal flush|7 cards) = probability of a royal flush given 7 cards = 4*(47 choose 2)/(52 choose 7) = 0.00003232
- RFX = p(royal flush|X players at the river) = probability of a royal flush with X players at the river = X*RF7-(X-1)*RF5

For RFX the (X-1)*RF5 factor removes the over-counting when the community cards contain a royal flush. The probability of that event is RF5 and it occurs once in each RF7. So subtracting X-1 of them leaves one for everyone to share. Plugging in the numbers for X = 10 yields 0.00030934 or 1 in 3,233 hands.

In reality, the odds are much lower as ten people staying in for the river card at the same table does not happen often, if ever. In many cases, the hand is over before the turn or river card gets shown and if the river does come into play, only a few people remain in the hand. So if on average, one out of every three hands goes to the river and when that happens, on average three players remain RFX is more like RF7-(3-1)*RF5 = 0.00002924 or 1 in 34,200 hands.

My other co-worker attacked the problem in a slightly different way, by summing the three probabilities of various ways a royal flush can occur in a game of Texas Hold’em i.e. 3 of the 5 community cards and both of cards from a single player make a royal flush, 4 of 5 community cards and one card from a single player make a royal flush and when 5 of 5 community cards and 0 cards from every player make a royal flush. To compute the odds that way, let’s set some variables (these again all presume all players at a table stay in until the river every game):

- C1 = p(1 royal flush card|royal flush, X players) = probability of a player getting the 1 card necessary to complete a royal flush given a royal flush is happening and X players are at the table = 2*X/(47 choose 1)
- C2 = p(2 RF cards|royal flush, X players) = probability of a player getting the 2 cards necessary to complete a royal flush given a royal flush is happening and X players are at the table = X/(47 choose 2)
- C3 = p(3 of 5|royal flush) = probability 3 of the 5 community cards are used by a player given a royal flush is happening = ((5 choose 3)*(47 choose 2))/(52 choose 5)
- C4 = p(4 of 5|royal flush) = probability 4 of the 5 community cards are used by a player given a royal flush is happening = ((5 choose 4)*(47 choose 1)/(52 choose 5)
- C5 = p(5 of 5|royal flush) = probability 5 of the 5 community cards are used by a player given a royal flush is happening = (5 choose 5)/(52 choose 5)
- RFX = p(royal flush|X players at the river) = probability of a royal flush with X players at the river = 4*((C1*C4)+(C2*C3)+C5)

A few notes about these variables:

- C1 and C2 presume a royal flush is possible i.e. the community cards have the 4 or 3 necessary cards and just give the odds of a player having the remaining 1 or 2 cards. That’s why 47 shows up in those equations, because 5 of the 52 cards in the deck are already accounted for as the community cards.
- C1 has 2 as a factor in the numerator because a player has 2 chances of getting the remaining 1 necessary card. In C2, the player must have both cards necessary, so there is no extra factor.
- In RFX 4 is a factor because there are 4 possible royal flushes.

Plugging in the numbers for X = 10 we get:

- C1*C4 = (2*10*(5 choose 4)*(47 choose 1))/((47 choose 1)*(52 choose 5)) = (20*(5 choose 4))/(52 choose 5) = 100/2,598,560 = 0.0000385
- C2*C3 = (10*(5 choose 3)*(47 choose 2))/((47 choose 2)*(52 choose 5)) = 10*10/2,598,560 = 0.0000385
- C5 = 1/2,598,560 = 0.000000385
- RFX = 4*(0.0000385+0.0000385+0.000000385) = 0.00030954 = 1 in 3,233 hands

That’s exactly the same result via the first formula, phew. One non-obvious tidbit this way of thinking highlights is the probability of getting a royal flush with 4 cards in the community and one card in your hand (C1*C4) is exactly the same as getting a royal flush with 3 cards in the community and 2 cards in your hand (C2*C3).

Now that all the hard work is done, estimating how often one might expect two royals flushes at a ten player table where everyone stays in until the river on every had would be 0.00030954^2 or 0.00000000958 or 1 in every 10,436,778 hands. If I played 3,900 hands/year as originally estimated, I could expect this event to happen once every 2,676 years. Using our more realistic earlier estimate that one out of every three hands sees a river card and when that happens, there are three players still in the game, a royal flush would happen 1 in 34,200 hands (0.00002924) and a back-to-back occurrence would be 0.000000000855 or 1 in every 1,169,590,643 hands. For some perspective, the odds of winning the MEGA Millions Lottery jackpot are almost 7 times better, approximately 1 in 176,000,000!

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