A Probabilistic Analysis of a Game of Dungeon Dice

Dungeon Dice is a board game that Parker Brothers produced in the 1970s. Each player is a prisoner in a dungeon staging an escape, and needs to roll dice to tunnel their way out. What series of events caused you to be jailed in the first place is left to the players' imaginations.

Playing this the other night, my friend Catherine rolled all different symbols (the best combination you can roll, earning you three cards toward your freedom) three times in one game. I became curious about the probability of this and decided to figure it out here.

On any given turn, you roll six dice. To figure out the probability of rolling all different symbols, imagine you are rolling one die at a time. The first die can be anything (6/6 probability), the second die needs to be anything except for the first symbol that was rolled (5/6 probability), the third needs to be anything except the first two (4/6 probability), and so on until the last die, which has a 1/6 chance of being different from all the others.

To find the chance of all of these happening in a row, we multiply those probabilities. Here's a screenshot of a program called Jupyter (powered by the Python programming language) I used for this:

So the odds are 1.54% of doing this on any given die roll. But what about the odds of doing this three times in one game? 

I believe the game lasted five rounds, so we want the probability of doing this at least three times in five rounds. A quick Google search turned up this explanation of how to calculate binomial distributions. It's simple:

It gets into Stats 400 series territory here, but fortunately the Python scipy library has some helper functions for this. This is the calculation of this happening exactly 3/5 times is:

Or 0.00356%. Adding in the probabilities of it happening 4 or 5 times only changes this to 0.00359%, or 1 in 27,850.

As a check of all this, it should be somewhat more likely than the chances of doing this three times out of three (whose probability is much easier to compute). Multiplying our 1.54% chance by itself three times yields 0.000367%, ~one tenth as likely as the chances of doing it 3 out of 5 times, so our result above seems sane.

Conclusion: Catherine is very good at Dungeon Dice.

The Perfect Pulk - MN Ski Pulk I

Yesterday, I took a pulk that I put together out for its virgin voyage on Mt. Hood. Below are details of the pulk construction, notes on how it performed, and some pics.

First off, the definition of "pulk", from dictionary.com:

A pulk (from Finnish pulkka)^[1] is a Scandinavian short, low-slung small toboggan used in sport or for transport, pulled by a dog or a skier. The name of the sport is pulka. The sled can be used to carry supplies such as a tent or food, or transport a child or other person.

I had rented a Kindershuttle pulk in December to haul gear for an overnight trip, and it was a huge failure. It flipped onto its side about every 15 yards for the 6 mile ski in that Greg and Anna and I were doing, and then again on the way out the next day. It was supposed to be for hauling kids, but a child riding in that thing would have been severely damaged within the first mile.

So, it was with somewhat deteriorated expectations that I started setting up my own pulk, using instructions from skipulk.com. I have to make an aside here and send huge thanks tto Ed Buffard for writing the excellent Pulk Book PDF and distributing it for free. I wound up following their instructions for the "MN Ski Pulk I" very closely. The book has full details, but here are the few key ingredients to give you a sense of the construction:

• 1 Paris Expedition Sled
• 2 six-foot long 1/2" diameter solid fiberglass pole (I would not have thought to order livestock fence posts for this, but they were perfect)
  
• 2 ten-inch pieces of 1/2" ID flexible braided vinyl tubing
• 4 hose clamps to secure the poles in the tubing
  
• 4 mend plates with 4 bolts each to attach the hose to the sled
• 1 strap to keep the poles crossed (important for increased control of the pulk)
• 2 3/8" copper t-fittings, with the arms of the T sawed off and the remainder expoxied to the ends of the poles (to create an attachment point for the harness)
• 2 quick-link biners to attach the poles to the harness
  

The trip results:

• The loaded pulk weighed 35 pounds (w/o poles + harness)
• It was relatively easy to loosen the hose clamps with a screwdriver to remove the poles for transporting in the car. A little Dr. Bronner's helped reduce friction.
  
• On the snow, the pulk was unflippable. I took it over berms, on side-slopes, down and up hills, hit trees with it, did 180 degree turns, fell down multiple times myself, and still could not capsize it. It's as impossible to flip as it is to not flip the Kindershuttle.
• The poles seem to have enough bend to them that they didn't seem to make the falls any more awkward or push me as I was stopping (unlike the Kindershuttle, which left me with bruises on both hips).
• I did not put fins on the pulk as the book recommends, and it doesn't seem like they'll be needed. The pulk tracked well on its own, even on decent side-slopes.
  
• The only terrain that I encountered that it was difficult on was icy uphills. It made skiing upwards at least 5x more difficult than it would have been w/o the pulk. Ideas of ways to mitigate: transfer weight from pulk to backpack for sustained inclines, add skins to skis for uphills.
  
• Unfortunately, the skiing yesterday around Barlow Pass on Mt. Hood (Oregon) was mostly icy uphills, so after an hour or two, we stashed the pulk and skied around without it.
  
• Overall, I am confident that this thing is ready for some prime-time winter ski camping. This is the sled that will take me around Crater Lake.
  

The pulk in action

The harness setup

The pole attachment point with hose clamps

The pulk fits into my Yakima SkyBox as if they were designed for each other (with enough extra room for about 4 pairs of skis and poles)

The load it was carrying - just about everything needed for a few days on the trail. The two summer squashes were used to add a bit of weight and simulate the things I didn't feel like gathering for the day trip.

Two questions for you, readers:

1. What should I name my pulk?
2. How should I decorate/accessorize it?
   

Updates from N and S Thailand

I've been behind so I'm devolving to list format updates to catch up:

• The 2nd week of volunteering in Singburi (3/23-27), we taught at an English camp. A French guy and I signed up to help teach "The Planets" and were put in a class in front of 30 or so ~10 year olds. After waiting a few minutes for the main teachers to show up, we realized that that was us. The first day was a little rocky, but we hit our stride and had an amazing time teaching. Unfortunately, the language barrier prevented me from telling the story of how, when we pointed Hubble in a direction that we thought was devoid of anything, the long exposure came back filled with thousands of points of light (each representing a new galaxy).
  
• I spent a day at an elephant sanctuary near Chiang Mai around 4/1. I would highly recommend this to anyone going to Thailand. Highlights included washing the elephants in the river and seeing aspects of their social relationships with each other.
  
• My last few days in N Thailand (4/2-4) I spent in a small town called Pai, which is a little like Eugene, OR. I rented a motorbike and cruised around the countryside and worked/stayed at an organic farm.
  
• Yesterday I made it down to Ko Phi Phi Don. Phi Phi is a tropical island paradise, and has been in a movie w/. I've done some kayaking, snorkeling, and rock climbing here. To escape the island's main downside last night (the drunken hordes of spring-breaking Europeans), I went to sleep early. Then I woke up early this morning and hikes around the woods, eventually stumbling on a few Thais who squat in tents on a ridge overlooking the Andaman sea. They offered me some pumpkin rice dish (accepted) and hits from their bong made of bamboo (declined). Tomorrow morning I think I'll go up there for the sunrise and bring some fruit as thanks.
  

The heat-lightning storms at night here are enthralling, better than watching movies on the big screen.

On Fri, I'll be taking a boat to Hat Rai Lay, where there are beaches, caves, and some world-renowned climbing.

The Eurokids are getting rowdy, time to go to bed.

PS More pics up on flickr

Chiang Mai

I'm passing a couple lazy days in Chiang Mai right now. I took the 9+ hour bus ride on Sat from Singburi after the volunteer program wrapped up. On Sunday I wandered around the moat-encircled old city of Chiang Mai, ate at an Indian place, a veg place, and dropped into an orphanage for some volunteering. I didn't know what to expect from the orphanage, but it turned out to be a great time. When I showed up, the kids (about 20 of them, between ~16 and 30 months) were sitting around a table and stopped whatever they were doing to stare at me for awhile. I tried asking one his name in Thai, but that illicited no response. I began to worry a bit that they would be too scared of me, but we soon switched rooms to a play room, many beach balls were unleashed, and I was swiftly appointed as a human jungle gym. We moved to the playground after awhile, and I juggled some clubs for them, which completely blew their 2-year old minds. An American family with a translator/guide was also visiting and the mom kept saying how much she wanted to take some of the kids home to Chicago with her, and something about understanding Angelina Jolie now.



After 2 weeks in Singburi, Chiang Mai seems very diverse, cosmopolitan, English- and vegetarian-friendly. I'm picking up a few history tidbits- it was founded in 1296 as the capital of the Lanna kingdom, some buildings date back to the 1200s, it was sacked by the Burmese in the 1700s (moat fail).

Oh, I also got a Thai massage yesterday (1 hour for $5). It turns out to be 50% assisted yoga (ex. the masseuse putting you in a body twist then putting her weight on you to add to the stretch) and 50% muay thai (Thai boxing, ex. punching you in the top of the head, putting you in a headlock and twisting your back, elbowing you right to the side of your groin). I like it.

Today I visited a temple and talked to a Buddhist monk for awhile as part of a program they call "monk chat". They practice Theravada Buddhism in Thailand, which is the oldest branch, which Mahayana came from (the monk compared the relationship between the two to that of Catholicism and Protestantism).

Tomorrow- overnight stay at an elephant sanctuary, then 5 more days in Chiang Mai or Pai, perhaps with a rock climbing day trip/lead-climbing class in there. Then, a week from today, I'll fly to Phuket in S Thailand and hit some islands, snorkeling, rock climbing, etc.

Singburi Week 1

A little background for those I didn't get a chance to talk to before I left- I managed a month of unpaid leave and decided to spend the in Thailand. The first half of the trip I'm volunteering in a rural area north of Bangkok (via the IVS) and the other half I'll be traveling around, hitting an elephant sanctuary, maybe some islands, rock climbing, national parks, whatever sounds good at the time. For the photo version of this blog, here's my pics.

Now the update- I met up with the volunteer group on Monday morning in Sing Buri, had an intro day that day, then Tues, Wed, and Thurs working in a school (mostly on tiling a classroom floor). The volunteer group of 9 consists of 4 people from France, 2 from (French speaking) Belgium, and 1 each from Toronto, Germany, and Portland (me). French tends to be the default language and it's fun to listen to that and pick up a little bit of that and Thai too. There are a couple other volunteer houses a few miles away, and we run into them from time to time. There's a bar by one of the volunteer houses that looks like a lemonade stand, but they serve Thai beer and whiskey, we've had a couple debaucherous nights there already.

The work in the school is a nice mix of manual labor and playing with the kids. The kids are super friendly (actually almost everyone is in Thailand) and though they have no language in common w/ any of the volunteers, we find ways to communicate using sign language, soccer players' names, and gibberish. Next week we'll be teaching them English, let me know if you know of games or other tools that are good for teaching when the students and teachers don't speak the same language. I've got a couple simple riddles in mind, juggling, and duck-duck-goose of course.
We've got this weekend off and I'm either going to head to a beach or a national park. I want to see some monkeys.

Rental Pants vs the Emerald Buddha

I arrived in Bangkok late last night and found my way to my hostel about 23 hours after leaving my house in PDX. It seems the political situation here is less than stable right now, but the protests seem like they'll be peaceful- http://www.bangkokpost.com/

Tonight I'll hit the juggling club in Bangkok and tomorrow I take off for the volunteer program in Singburi. I'll try to get some pics up before then as I may be offline for a couple weeks.

For my fellow map aficionados- right now I'm approximately here.

Day Seven

I rode my first century today- did 105 miles from Lake Quinalt to the train station in Olympia. I rode through Hoquiam, tires for sale in a front lawn, Aberdeen with it's agro drivers, through Olympia past state capital building, and finally to the train station outside town. There were half a dozen retired folks in the train station who seem to hang out there as a hobby, talking about which signal lights are the most beautiful, discussing different types of trains, tracks, and movies about trains, and getting totally pumped up when trains go by. I wanted to stay and hang out with them.

Wildlife list for the trip: banana slugs, black banana slugs, roosevelt elk, barred owl, woodpecker, loons, deer, rabbits, red tailed hawk.

More pics from the trip