Posted by dwheatley
Jul 26, 2012

Rock Paper Scissors and Pascal's Pyramid

Cool Math Post Alert:

This is an amusingly tricky question to answer. After some messing around in Excel, it turns out the probability of tying after n games of RPS can be found from the trinomial expansion, whose coefficients are illustrated on Pascal's Pyramid.

The sum of the middle column in level n gives the number of ways to tie after n games.

You can find this in excel by adding up the previous 3 nearby numbers. I get 8953 for n=10, out of 3^10 = 59049 outcomes, probability 16.5%

Or, if you are into these really cool math artifacts, by doing 111^n, or 10101^n, or 1001001^n, etc.

10101^5 = 105153045514530150501

See how the numbers repeat on either side? Those are the sums of the columns, the number of ways of having any number of points after 5 games. The wiki page shows how to divide it into:

1 5 15 30 45 51 45 30 15 5 1

There are 51 ways to tie after 5 games.


Posted by dwheatley
Jul 25, 2012

Ultimate Texas Hold'em (UTH) strategy addendum

I am just making a note that Stephen's great UTH strategy has an 'error'. I am lifting this from a thread on this site.

"The situation is, I have a pair of 4s, one of the 4s is in my hole cards. Shouldnt I raise?"

"You don't raise bottom pairs against scary boards, like straight or flush draws.

I don't think Stephen caught the straight problem in his strategy (he knows about not raising against flush draws), but I've noticed it from playing too many practice rounds with the calculator on. I plugged your hand into the wiz's calculator, and it confirms you should check. It's borderline, -0.584 vs -0.574."


odiousgambit Jul 25, 2012

I've been playing without too many errors at the Wizard's "trainer", developing an instinct at this game.

dwheatley Jul 25, 2012

me too. One I routinely forget, but is in the strategy, is to raise a gutshot straight draw with top kicker. Most recent hand was Qx and the flop comes AKT. I instinctively checked, but the correct play is raise.

Hunterhill Jul 25, 2012

Not sure what your exact hand was,but the board has to be a 3 card open ended straight draw to check,if the straight has a gap then you should raise.This assumes there is no flush draws. Stephen`s strategy and Grosjeans are not perfect they have to make some concessions so that the strategy is usable.With a 5 or higher in your hand and it pairs up on the flop you would raise even with a 3 card open ended straight showing.

Wizard Jul 25, 2012

In my opinion, we can pretty much throw out Stephen's and JB's UTHE strategies because the one by James Grosjean is significantly simpler and better.

dwheatley Jul 26, 2012

The hand in question was 4h 9s,

flop was 4c 5d 7c.

Note the straight has a gap, but also has 2 suited cards not matching the hole cards. This is a checking hand. If we change the 7c to a suit that matches the hole cards, it becomes a raising hand.

As for JG's strategy card, I'm sure it's good, but it's not free.

Hunterhill Jul 26, 2012

You can buy JG`s card for 5.95 so it might pay for itself after a few trips. There are so many minor exceptions that it would take some serious study to learn them all.

Hunterhill Jul 26, 2012

I also just noticed if you change your 9 to a jack or higher then it becomes a raise even with the 2 suited cards on board.

teddys Jul 28, 2012

The Wizard showed me Grosjean's strategy card for about 15 seconds but it did look simpler and much shorter than any other strategy. He claims it performs at a 2.1% house edge. I spent way too much time learning dg.net's strategy, and now I just use the Wizard game. I think I play pretty close to optimally.

Posted by dwheatley
Jun 23, 2012

Lucky Lucky at my local casino

I discovered 3 BJ tables with the Lucky Lucky sidebet at my local casino. I don't normally play BJ anymore, focusing my +EV efforts on Spanish21 instead (a lot less heat, less angry patrons, generally more fun to play).

However, having recently seen it on discountgambling and here, and remembering it described as 'crushable', i learned a new count in the bathroom and gave it a go. The game was 8 deck, S17, DAS, no surrender. As I recall, that's 0.43%, which i can achieve on auto-pilot. That part would prove to be important, because the new count was a bit of pain to keep up with.

First shoe I entered mid-shoe and started practicing. The main dealer was young, competent and quick, which caused me to lose my practice count a few times. One of the guys is playing lucky lucky for $20 a hand, every hand. Near the end of shoe, with what i think is a negative count (I have no idea what the real count is), BOOM, he hits 777. $1000. I start drooling as I imagine my quarters (max $25 bet) out on that bet when the count is good.

During the break I figure: at $10 a hand, I was giving up $0.04 a hand on the game. Discountgambling suggests I will get a good count 22% of the time, with an average advantage of 4%. If I put out $25 each good count, that's $1 EV 22% of the hands. I am looking at a net profit of $0.18 per hand.

First shoe turns negative for most of the shoe, until it swings positive for the last 3 hands. Lose, hit 21 for 3to1, lose. I am up $25 on the sidebet, and mysteriously seem to be even in blackjack. I'm not even paying attention to the BJ game.

Second shoe goes very negative, and never recovers. I also lose ~$100 due to a dealer who can't bust.

Third shoe goes very positive. Very positive, running count of +30 with 6 decks to go. I had $25 out there for 2/3 of the shoe. I was rewarded with... bad variance. Dealer still can't bust, and also can't deal me 7s. I hit a 20 and 21 for 2 or 3to1 a couple times, but overall its a minor disaster. Down another $200, and I check my meager bankroll and decide it can't handle another shoe like that.

I'll get 'em next time.

I practiced with my deck at home the next day. I swear every time the count went positive, I dealt myself a winning sidebet hand. I am plotting my revenge.


buzzpaff Jun 23, 2012

Gee.,losing in a casino and then winning at home. Has that ever happened to anyone else ?

Posted by dwheatley
Nov 10, 2011

Hawaii vacation - part 2

Other Activities

Pearl Harbor

Everyone in my family wanted to go to Pearl Harbor, so we stopped in on the way north (I keep typing it as harbour, but I'll respect the American spelling here). The wizard's review is extensive, I will add that a lot of it is free, such as the museum and trip to the USS Arizona. Unfortunately, the navy canceled the boat trip to the Arizona due to inclement weather. It didn't seem so bad, but what can you do...

We got there at 9:30 (on a Tuesday I think) and we were given a 10:45 ticket for the movie, which was just enough time to see the museum. We chatted with some family members of vets, overall a good experience. Don't miss.

Diamond Head

We also climbed Diamond Head. A short but steep climb along somewhat rocky terrain. Fee to park is $5 per car, or $1 per person to walk in. The hike from outside to the parking lot looked boring, I would park, even with a small crowd.

At the summit, there was a park ranger type (maybe not actually a ranger) hawking tours. There were dozens of people, pretty crowded. The hike is not very well shaded, so it got hot. Most people can make it, given enough time. Easy enough to jog if you are in good shape.

Dole Pineapple Pavilion

We did the maze. I likes mazes. No, I love mazes. My sister and I traveled together, almost never splitting up, and got all 8 stations in 50 minutes. The standing record was 8 minutes. What? You would have to know exactly where they all were, and probably run through the holes in the maze. They have tried to repair some, but others are almost too tempting to ignore. 8 minutes... no way.

It helped knowing the expansion only had 2 stations. The maze does cost money ($6) I think, but there was no attendant when we showed up. So... free maze! There's a coupon in a book somewhere for 2 for 1 admission, don't miss that. Or the maze.

Waiamea Valley Park

Opposite Waiamea beach is this gorgeous park we nicknamed Jurassic Park, as I sweat parts of it were shot there. It is an archeological site for an ancient Hawaiian village, transformed into a botanical garden and swimming hole. $15 per person to enter, half price children and seniors (60+).

There is as expansive garden featuring tropical plants from all over the world. Over 1000 type of trees, plants and flowers, mostly leafy plants. It is about 3/4 a mile long, with multiple side paths. Easily could take 2-3 hours to wander the paths alone.

Along the valley there are remnants of the village, with information signs and the occasional guide to teach you more about the ancient way of life and culture. That was pretty interesting, but the huge sprawling plants clearly dominate the place.

At the top is a waterfall + pond you can swim in, no extra charge. Bring your bathing suits! The lifeguard gives you a brief safety warning (the undertow can be strong, don't actually swim in the waterfall for fear of getting bonked on the head by logs, etc.) and off you go. The water was cold and reddish brown. Cold because it's a river (but not cold by Canadian standards, maybe ~70 degrees), reddish-brown because of the iron-oxide present in the ground all along the north shore. It was actually closed for swimming the day before due to heavy rains, so time your trip. On the other hand, I have read that if it doesn't rain for a while, the waterfall turns clear. The current was strong, it would be very difficult to physically get to the waterfall at the rate it was moving water. Overall, fun times.


Nui's Thai food exists! It is in the middle of nowhere, and is as delicious as the wizard made it sound. There are TWO picnic tables, under a tent. Moving on up!

For other food options, I strongly recommend finding some Ahi Poke at a grocery store like Foodland, or some of the chuck wagons. It is Ahi tuna sashimi, mixed with some garlic, or onions, or sauce, to make a sort of sashimi salad. If you like tuna, dare to try the Poke (pronounced pokey, I think it refers to the 'salad' style, because you can get mussel poke).

Track down the Impossibles Pizza wagon on the north shore. The pizza is one size only, and doesn't fit in the box. Wowzers. Try Kalua Pig, it's some kind of super bacon ball on a pizza. I heard that the wagon is only allowed to stay in one place for about ~30 minutes, but I think it cheats. They will also tell you a custom pizza will take a long time, in order to get you to order a pepperoni or cheese pizza. Do not be deceived! We hummed and hawed, ordered a custom anyway, and were told it would be 20 minutes. Secret's out.

Posted by dwheatley
Nov 08, 2011

Hawaii vacation - part 1

I went to the island of Oahu in Hawaii (Oct 30 - Nov 7) for a destination wedding. The' rel='nofollow' target='_blank'>http://wizardofodds.com/news/Oahu.html]The wizard's review of Oahu was very useful for planning my trip, so I thought I would give my own brief review as an addendum. Hope you find it interesting.

I stayed with my family in Waikiki for two nights, then on the north shore for 5 nights. I stayed further north-east than the wizard did, near Waiamea bay, in a beach house. Great for families.

The weather was unpredictable. Clouds would form out of the blue sky, then rain on us for brief periods (5-20 minutes), then disappear. Others would roll in over the mountains without any morning. When out, the sun was hot. Overall, a great experience.

Beaches & Surfing

The Kamehameha Highway that runs from Haleiwa north-east through Waiamea is a great spot for beaches and surfing. I nicknamed it surfer's highway. The highway was getting very crowded by the end of my stay, as many surfers were coming in to town for a big competition. I was told that a year or two ago 20,000 people crowded the north shore to watch the 30+ foot waves and surfers. When I was there, the surfers were riding ~10 foot waves

Ke Iki

This is where the wedding was. It's just north of Pupukea Beach Park. Bad shorebreak, thus hard to swim or boogie board. Great for a long stroll and watching big waves slam the beach. The people least afraid of shore break are 10 year old Hawaiian girls. Unbelievable.

Shark's Cove

Shark's Cove is not so much a beach as good snorkeling grounds. Conditions were not great when I went. Lots of waves and a little murky. Almost dangerous at times. I'll trust the wizard did indeed have a good time, but I had much better snorkeling at Hanauma Bay Beach Park.

Waiamea Bay

Very pretty horseshoe beach with mountains rising on either side. Better for swimming, but the shorebreak can still be rough. My uncle sprained his hamstring and was carried off the beach in a stretcher. He's ok now. Kids do jump off the rocks at the one end of the beach, that's fun to watch but crazy to try.


I went surfing at Haleiwa twice. The first was for a lesson with Uncle Bryan's company, called Sunset Surratt Surf School. My sister and I had a semi-private lesson (not specifically requested) that lasted over 2 hours for $80 each. This included the boards & rash guards (special tight shirts you wear to prevent sun burn an board rash). Do NOT go surfing without a rash guard. My cousins are living testament to the owwww.

The surf lesson is great for a number of reasons. They give you beach training so you know what to do when you are there. Then the guy tows you around with his toe on your board (he 'toes' you around?) so you don't have to much paddling. Paddling is hard, and your arms will turn to jelly if you don't have training or breaks. Once you are out there, he tells you which waves to go for, then pushes you into them, then coaches you when to stand up. You will catch waves, and ride them a long way if you get a good lesson. If you don't... good luck!

We went out a second time a few days later with just a board rental (~$20 per person, including rash guard). It was tiring, but the lessons worked. My sister and I both did great without getting pushed. So much fun.

Hanauma Bay Beach Park

Best snorkeling ever. Huge fish (2 - 3 feet), some big schools, ok coral. I found a sea turtle (!) eating the reef near the ledge out to sea. Awesome. Downsides: $7.50 per person, plus rentals. Rentals cost $5 (mask) - $12 (full set with flippers), depending on what you want. Deposit required, so you need a credit card or extra cash ($30 I think). Lockers are $7. And you need to watch a weird, boring at times, amusing at others, information video. That takes 9 minutes, only running every 15. It took us over 30 minutes to get to the beach. Once there: awesome.


Waikiki is crowded. Lots of places to rent stuff. Easiest place to swim that I went to.


Ayecarumba Nov 09, 2011

Thanks for this dwheatley. I enjoyed it. Are there lockers at all the beaches? I always wondered what the surfers did with their cell phones and car alarm key bobs while they are out on the water so far from the beach.

Did you get a chance to tour the USS Missouri at Pearl Harbor? I highly recommend it to everyone who is going to Oahu.

1BB Nov 09, 2011

Your great report has me thinking about planning a trip. I like Oahu and Waikiki in particular and my wife loves Maui. Whenever we visit we split our time between the two.

Nareed Nov 09, 2011

Did you get any coffee?

Wizard Nov 09, 2011

I enjoyed your report and thanks for mentioning mine. The conditions were wonderful when I went to Sharks Cove. I never saw any lockers at the state beaches. What surfers probably do is leave all their valuable stuff in their and just carry a car key on them. Next time I go to Oahu I'll follow your advice and rent a surf board. I had a surfing lesson Santa Barbara a couple years ago, hopefully that will help.

dwheatley Nov 10, 2011

Surfers don't have valuables. They have surf boards, which come with them. I am pretty serious about that.

More seriously... A lot of people come in groups, and leave spotters on the beach. This iss good for taking pictures, guarding valuables, and calling the lifeguards if something goes wrong.

I didn't see any lockers anywhere except for Hanauma (which is pronounced BOTH ways: Ha-na-u-ma or Ha-now-ma, the locals can't seem to decide. Most call it Ha-now-ma)

dwheatley Nov 10, 2011

About surfing at Haleiwa, this particular cove was very nice for beginner surfing, right at the north end of the beach. There is a jetty on the right side. All surfing areas have places where the waves come in, and riptides where the water goes out. Smart surfers learn these areas, and paddle out on the riptides. This saves energy and gets you out of the way of the surfers. It is much better to paddle on an angle out to the riptide, then turn to the ocean to get back to your spot to catch waves. You know you found it if the waves don't seem to crash and you are moving out without paddling. At haleiwa, it goes to the right along the jetty. Easy to find.

Also, we saw sea turtles there. Taking a break to watch sea turtles from your board is good surfing.