December 19th, 2012

Anniversary Of A Sort

This is not a post about sorting algorithms; nor is it celebratory. December 10, 2012 marked the 1 year anniversary of what seems to be the worst event in my life to date (and perhaps some of the other people involved). I have resisted writing about it in the past because I felt the need to say too much and there were too many loose ends, but perhaps a complete revolution around the sun has provided enough perspective to get this post mostly right.

Clearly you’ll find my experience pales in comparison to countless horrific events in human history and many people might rightly believe I live a charmed life; but relative to all my other experiences, it and the subsequent events that followed constituted a global minimum, the major low-point in my life. For reference it trumps my parents getting divorced, my lola dying, getting a D in a 5 credit college course etc. Those events, while extremely unpleasant, differed in two important ways:

  • They did not come as a surprise
  • For the most part I had no control over them

Ok, I’m sure I could have worked harder during the college course but the professor and my TA were useless and nothing in my life up to and including that point helped me understand why the Cauchy-Schwarz inequality or Fourier transforms were useful :)

On Saturday December 10, 2011 at 6:33am a phone call awoke me, which I ignored as the phone was in another room. For several years while working at I kept the phone by my bed as I was the primary contact for the Amazon MP3 portion of the website. Phone calls at all hours of the night were expected and needed to be responded to immediately. However, when the phone rang a second time a minute later, I got up to see what was going on. I did not get to it in time to answer, but noticed the phone number belonged to one of our neighbors at our loft.

When we moved to Berkeley at the end of 2009, and tried to sell our loft for almost a year, the housing market was in a state where the offers received did not make financial sense to accept. So eventually I broke down and became a landlord. Luckily our renter turned out to be a perfectly normal person and our real estate agent provided property management services at an extremely reasonable price such that the whole landlord experience, while certainly not enjoyable, did not in itself become THE low-point, just merely a stop on the way.

The voicemail was 37 seconds long and explained there was a fire in the loft unit next to ours, the fire seemed to have been put out but there was smoke everywhere, the fire department was going into our unit now and our tenant was out safely in the courtyard. We headed over to the loft as soon as we could and spent the next 8 hours speaking with our tenant, other residents, the fire department/inspector, our homeowners association property manager, a representative from a reconstruction company and one from a cleanup/restoration company as well as cleaning out/drying off certain items from our loft. I tend to try and do everything myself e.g. my wife and I attempted to vacuum water up from a 9’x14′ rug, but after a few hours, realized we weren’t making much of a dent in the cleanup effort. Everything smelled like smoke, we had no way to transport or store all the items and decided to let one of the cleanup/restoration companies handle things. Thus began a strange 9 month journey.

Good times

Had the fire taken place at our house, in almost all respects life would have been worse; however, owning the loft itself was already a large thorn in my side and this event felt like someone replaced that thorn with a salt tipped, hot, rusty sword. Every interaction I had, no matter how nice and/or reasonable the other party tried to be, felt like torture. And the exchanges with completely unreasonable, incompetent people were of course orders of magnitude worse.

Breaking the lease with our tenant and refunding their security deposit was pretty straightforward. What the fire didn’t destroy, the water from the fire department’s hoses did. I met with the representatives of the cleanup/restoration company the next afternoon and they moved out all the items we planned on keeping, would clean/ozone them and store them until the loft reconstruction was completed. Over the course of the next week I met with our insurance adjuster, the homeowners association’s adjuster, board of directors and property manager, a fire inspector from our insurance agency, a City of Oakland fire inspector, a construction manager from the company performing the repairs on the behalf of the homeowners association and one on behalf of my insurance company. After the initial shock of the fire, my spirits rebounded fairly quickly as it seemed like there might just be the initial up-front cost of getting everything in order, 3 months later everything would be repaired, and we could enjoy the trip to Cambodia we had planned for New Year’s.

That rare feeling of optimism slowly turned to despair as the repair dragged on and I spent time every week on the phone with various parties coordinating; sometimes I had to be on site at the loft. Some of the stepping-stones (hard-knock lessons) to the bottom were:

  • Make sure to have landlords insurance when renting a space to another person. Your existing homeowners insurance isn’t good enough and won’t cover lost rent.
  • Be very particular about every single line item of work e.g. “ozoning and smoke abatement” sounds great after a fire until it turns out that part of smoke abatement involves the contractor spraying white sealant on your floor instead of a clear one because they assumed you were going to put a new floor over it.
  • Check on the status of your material delivery right up to the point where you leave work to meet them on site. Don’t expect them to call and cancel/postpone, they’ll likely just not show up.
  • Ask your real estate agent to contact the buyer’s agent and title company representative at least once a week to check in on things and get specific details regarding how the loan funding and other related process e.g. homeowners association documentation is going. Don’t settle for generic statements like “the underwriters are still looking at it”, you want something more along the lines of “the lender is requesting more asset history as the buyer’s income has been volatile as of late”.
  • Don’t be afraid to ask for compensation if the buyer delay’s the sale. Some penalties are already built into most contracts, but you can make them up as you go along as well e.g. if the buyer pushes the closing date out one month you can ask them to cover the extra month of mortgage, property tax, insurance etc. The worst that can happen is they say no. If you’re worried about them getting offended and backing out completely, that would likely trigger a steeper penalty built into the contract so it really isn’t an option.
  • If you live in a building with a homeowners association, read the governing documents thoroughly and if you don’t understand them, ask a board member or management agent or lawyer to explain. Be very vigilant about what goes on during the board meetings. Only run for a seat on the board if you desire to get wrapped up in the minutia of other resident’s lives and expose yourself to lawsuits.
  • Proactively cancel any EFT/ACH even if there’s a penalty. Don’t rely on the other party getting it right in time.

The construction finished in May and after a bit more cleaning, our unit went on the market in June. We priced it “low” at $169,900 (we purchased it for $268k in 2002 but the most recent sale for a comparable unit in 2011 was $114k [short sale]) in an attempt to entice offers. It worked as we received 3: $170k (25% down), $175k (10% down), $180k (20% down). The buyer who offered $170k was looking at it as an investment property, the buyer who offered $175k wrote us a nice letter and intended to live there and the buyers who offered $180k had a lot of demands and extra contingencies. After some negotiation and clarification, it turned out the buyer who had written the letter actually had 20% down and was willing to match the $180k offer and we decided to accept that one. Then at the last minute, someone came in an offered $220k. Figuring it was too good to be true, but with not much to lose and a whole lot to gain we entertained it but it fell through over concerns relating to the fire.

Getting rid of some paint

After clearing a few more unexpected hurdles e.g. delay in funding of the loan and the discovery that our dishwasher wasn’t working, our unit closed escrow on September 4, 2012, saga over right? Not quite, I fielded verbal threats and agitated phone calls from a few parties related to the payment of the reconstruction and had to get refunded for overpaying my homeowners association dues due to EFT/ACH timing and close of escrow notification to the accounting company. To this day I’m still dealing with various departments in the county of Alameda and city of Oakland that don’t seem to have missed the memo stating I am no longer responsible for property taxes and have no rental business income to tax. Come February/March I have an exciting date with H&R TaxCut Premium edition and the intricacies of section 1231 property. Hopefully the government will be kind enough to help us recover some of the losses from the rental/sale of our loft…if they don’t, it’s water under the bridge at this point; and the last 3 months of not having to worry about the loft and knowing that it’s unlikely we’ll ever have to concern ourselves with it again is worth the price. At the end of the day, I’m just thankful to have been in a position to purchase my own freedom.

July 27th, 2012


As a kid I often wanted a pet [dog] that I could play with, but my parents weren’t too interested in having any (we had some freshwater fish) and I likely couldn’t handle the responsibility myself; so I went pet-less until around the age of 11. At that time, we reached an agreement that I was capable of caring for a hamster. After an eventful trip home from the pet store (Huey, named after the hamster in the Calvin and Hobbes comic) chewed through the cardboard box in the car, climbed up into the ventilation system and had to be lured out via a corn kernel), I managed to care for Huey until he died of natural causes a little over 2 years later.


He contracted some sort of respiratory illness, becoming lethargic and wheezing; my parents humored me by taking him to the vet. The vet in turn claimed to have put him in an oxygen chamber, but Huey died a few minutes later (perhaps the vet actually gave him a lethal injection, I’ll never know). Much like razors and printers, the business model of owning a hamster revolves around an initial cost e.g. razor handle, printer, hamster cage and then smaller continuous expenses e.g. razors and shaving cream, ink/toner and paper, hamster and food. So for a few dollars I picked up another hamster, Andiamo and he lived for a bit more than a year before I found him one morning somewhere between Algor mortis and Rigor mortis. By now I was 15, living in a townhouse with my mom and convinced her I could take care of a dog.

My idea of a dog lay closer to the dog’s ancestors e.g. wolf, coyote so only a few breeds would do. I was quite active (ran, rode my bike/rollerblades etc.) and decided I wanted a Siberian Husky. We visited a Husky rescue group and after several meetings ended up with adopting one. I named him Weiland.


Unfortunately, I was not ready to take nearly full responsibility of the dog. Huskies shed a lot and no matter how often I brushed him, I still ended up with more vacuuming duties than I wanted. As we didn’t have a yard and my mom worked full time, while I was at school he was in a crate (we tried to keep him in the kitchen but he could jump over any barrier we setup). And I just couldn’t consistently wake up early enough to take him for long walks before school. After a few months, we ended up giving him away to a family that had a husky and a yard and was looking for a second one (they’re very gregarious).

Back to hamsters, I went through Pierre and Surimi over the next 3 years, took a break when I went off to college and then during my junior year picked up General Nogi. The General was a badass, lived for almost 3 years and survived a trip down a flight of stairs in a hamster ball. When I moved into an apartment after college he caused the people below me grief, running around in his ball on a hardwood floor. Once he died, I didn’t get another hamster until I moved a year later. Fuzzhead managed to escape once but was successfully lured out of hiding with strategically placed piles of food. He also developed a similar respiratory condition as my first hamster, Huey, but recovered after we moved his cage to a different spot in the loft (presumably with less dust). After he died, I went pet-less for the next 5+ years.


My wife’s family has had many pets, birds, rabbits, cats, dogs, fish, hamsters etc. I’ve known her for 14 years, during many of which they had 3 outdoor cats (Sam, Mugsy and Wilbur) and until last month they always had a dog, Maxwell. We had discussed getting a dog when we moved into our loft 10 years ago, but decided it wouldn’t be fair to the dog since there wasn’t a yard and only a few patches of grass in our neighborhood. We also couldn’t agree on what type of dog to get as she preferred a dog that would be comfortable to have on her lap and I one that I could go hiking with, could run faster than me (I think I can still beat a dog with legs < 1 foot in length in a straight race, I certainly was able to run faster than Maxwell [Yorkshire Terrier]) and looked like a "real" dog i.e. closer to wolf/coyote than not. In addition, there were the usual logistics e.g. we enjoyed travelling and both worked full time. A couple of years ago we bought a house with an enclosed backyard, removing one of the primary obstacles to owning a dog. My wife had also been working at a job for the last few years, mainly in the afternoons/early evenings which meant a dog wouldn’t be home alone for 10+ hours/day. So the decision really boiled down to what kind of dog to get and what to do with it when we went on vacation.

Another change in our behavior when my wife changed jobs, was a reduction in frequent short 3 or 4 day trips (unless it was a holiday weekend since she no longer had vacation days), and a focus on longer trips when she had multiple days off. In-between the less frequent, longer travels my wife would go through periodic bursts of Craigslist and Petfinder searches; every now and then we’d visit a shelter or dog adoption day, but could never agree. I still felt the desire for a dog with roughly the size and look of a coyote, but that would make it difficult for my wife to pickup and hold or have on her lap for extended periods of time.

After a trip to Cambodia at the end of last year, we had another lull before our next batch of travel plans went into full effect (annular eclipse in New Mexico, Yankee games in New York, weddings in North Carolina and Colorado). After a disappointing trip to the Berkeley SPCA and a few more dog adoption days we happend upon a group of rescued dogs at a Petsmart in Concord. There was a curious reddish-copper dog on top of a crate, curled up in a ball alongside a cat, both sleeping. All the other dogs were barking and jumping in the makeshift pens; every now and then the cat or dog would stand up to adjust positions and then settle back into their naps. We spoke with the woman who ran the rescue about the dog for a little bit, Gina held her and we walked her around the store for a little bit. She appeared to be a mix between a chihuahua and a dachshund, was still a puppy but nearly full grown (a little underweight). She remained very calm amidst all the people, looked like a “real” dog and would be a good size to hold on one’s lap (10 pounds), so we signed up to adopt her. She had yet to be spayed or microchiped and already had an appointment at the vet the following week, so we put down a deposit, filled out the paperwork and came back the following Saturday.

Rescuing Annabellatrix

The car ride home was uneventful, she laid down in the crate we had bought. She still had suchers from the surgery and slept a lot the first few days we had her in the house. We took her out in the back yard at regular intervals in case she had to go to the bathroom and quickly adapted to going in an area along the fence which we rarely had reason to walk through. Amazingly, she has yet to have an accident in the house or the car. Her behavior has been better than could possibly be expected, she gets along well with all dogs; even ones that don’t like other dogs seem to like or at least tolerate her.

This isn’t to say everything has come up roses; she seems to be intent on collecting diseases, fungi and parasites. Within the first few days, her incision became infected causing her to go on antibiotics. With a weak immune system, she was unable to fight off ringworm which she either already had from the rescue or contracted somewhere in our backyard. A few weeks later she ended up with a tapeworm and most recently contracted giardia. None of these seem to phase her and only the ringworm was really an inconvenience for us.

She is able to (videos) fetch and loves to run crazy laps in the backyard. If you’d like, you can follow her exploits: Annabellatrix on Facebook and Annabellatrix on Flickr.

March 23rd, 2012

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!