What are the Odds?
Randy Evans
United States Richmond Virginia

I've been playing games most of my life. From time to time, my curiosity just gets the better of me, and I just have to know the probability of a certain gaming outcome. Most recently, this happened with the probability of each ops point total in a starting hand of Twilight Struggle, but this is not the first quixotic probability quest I've taken.
This list documents the questions and answers I've discovered along the way. Feel free to add your own probability questions, whether answered or not, at the bottom. If I have the time, I'll try to find the answer.

Randy Evans
United States Richmond Virginia

"Will I land on Free Parking this time?"
It all started with Monopoly when I was about 8. What were the chances that you would land on "Free Parking" when you were "Just Visiting" Jail? This happened fairly frequently, and somehow I wondered just how likely rolling that 10 was.
Somewhere in my wonderings, my older brother informed me that a "7" was the most likely. I figured that he was lying (just like he did about the advisability of trading Baltic for Park Place). Somehow I had thought 10 was the most frequent roll because it came up a lot in Monopoly.
Of course, I found out much later that the chance of rolling a 10 with 2d6 is Spoiler (click to reveal) 1/12 or ~8.3%. It seemed to happen a lot more frequently than that, particularly when my sister was rolling.
Dice probabilities for those who are interested. http://en.wikipedia.org/wiki/Dice


2.
Board Game: Risk
[Average Rating:5.58 Overall Rank:15220]
Randy Evans
United States Richmond Virginia

"Who is favored with three dice attacking and two defending?"
I started wondering this at around 10 when we started playing Risk (because my family was reaaaally sick of playing Monopoly with me). It seemed like the attackers were favored, but I wasn't sure.
I finally sat down and figured this out by hand when I was studying for the Bar Exam (because I was reaaaaally sick of studying legal mumbo jumbo). I remember working out what I thought were all the possibilities and coming to the conclusion that the attackers were favored. This is correct, but there are 7776 possible outcomes. I was bored, but I wasn't that bored. I think I used incorrect reasoning to come to the right conclusion.
More recently, with the aid of a spreadsheet, I figured this out. The attacker enjoys a healthy, but not overwhelming, advantage.
Spoiler (click to reveal) 37.17%  Defender loses 2 33.58%  Both lose 1 29.26%  Attacker loses 2


Randy Evans
United States Richmond Virginia

"Which is the easiest number to win?"
In my late 20's, I started playing games again and discovered a gaming renaissance had occurred. In addition to the new Eurogames, I discovered the old classics by Sid Sackson as well. I loved this game, but I always wondered what was statistically the "easiest" number. Was it easier to get three 2's or thirteen 7's?
It's a pretty complicated question because it all depends how you choose the dice in the game. I finally simplified the question by asking, "If I rolled 2d6 repeatedly, which number would win on the Can't Stop board?" This was still complicated to calculate, so I just wrote a computer program to simulate 1 million games.
Spoiler (click to reveal) The answer: 7 wins (no surprises)
Note: This lead to my six year old daughter to object that it was not fair that 7 had the advantage. She felt particularly bad for 2 (but was relatively untroubled by 12). I had to create race game for her where 7 was not the favorite.


4.
Board Game: Poker
[Average Rating:6.70 Overall Rank:884]
Randy Evans
United States Richmond Virginia

"What are the odds of each hand?"
OK, so this is pretty easy to look up, but in my late 30's, I became a math teacher. Hey, I could actually figure this stuff out, so I did. It was fun just because some are rather tricky. It is also very easy to make a mistake even if you do the math right.
For example, Spoiler (click to reveal) I was flummoxed trying to calculate the number of possible Flushes only to finally realize that I had to remove all of the Straight Flushes (which had already been counted).
Poker hands with odds explanations http://en.wikipedia.org/wiki/Poker_probability


Randy Evans
United States Richmond Virginia

"What is the probability of each possible ops point total in a starting hand?"
This, my most recent windmill, was a giggle, particularly since I included the China Card the first time in my calculations and had to redo them. All.
It was also the inspiration for this list.
Here are the results:
Spoiler (click to reveal) 5  0.000043% 6  0.000902% 7  0.007063% 8  0.04% 9  0.13% 10  0.39% 11  0.97% 12  2.06% 13  3.81% 14  6.19% 15  8.91% 16  11.46% 17  13.18% 18  13.57% 19  12.48% 20  10.23% 21  7.42% 22  4.72% 23  2.59% 24  1.21% 25  0.47% 26  0.14% 27  0.03% 28  0.00472% 29  0.00034%
You can go here if you want the whole sordid story. http://www.boardgamegeek.com/article/11554662#11554662


6.
Board Game: Dominion
[Average Rating:7.66 Overall Rank:73]
[Average Rating:7.66 Unranked]


A fairly easy one to start with (although I might jump into some more fiddly scenarios should I find the time).
Probability of opening hand containing: 2 Copper, 3 Estates = 1/12 = 8.33% 3 Copper, 2 Estates = 5/12 = 41.67% 4 Copper, 1 Estate = 5/12 = 41.67% 5 Copper, 0 Estates = 1/12 = 8.33%


Randy Evans
United States Richmond Virginia

"What is the chance that you will roll a yahtzee?"
I've wondered this one for a long time. Of course, it is not that hard to figure out what the chance is that you will roll a yahtzee on the first roll. That is 1/1296 or 0.077 %. (Pretty unlikely. If it were a poker hand, it would beat a full house.)
But you get three rolls in yahtzee, and you get to keep some dice as you go. Pretty complicated problem and frankly beyond my math capabilities. I was going to write a simulation to Monte Carlo the answer, but I never quite got around to it.
The answer:
Spoiler (click to reveal) 4.6% Not bad. You're about even money to get a yahtzee every 15 rolls.
For a lovely explanation, check out http://www.datagenetics.com/blog/january42012/index.html



