Pandemic: The Cure - standard deviation and probability

Initial setup

Recently we have been playing Pandemic: The Cure. The goal of the game is (loosely) to cure all the diseases. Each player has a certain number of dice that they roll on their turn (5 for most players, 7 if you're the Generalist) that give them their possible actions for the turn. One of the actions lets you use one of your die to "bottle up" a disease die and at the end of your turn you roll your bottled up disease dice and if the total amount rolled on the dice for a particular color of disease is greater than 13, then you have cured that disease.

Bottling up the diseases is great because it removes that disease die from play and helps you discover the cure, but until you discover the cure that die of yours you used to "bottle" it up is locked up and you can't use it, meaning you'll have fewer possible actions on your turn, making you less effective until the cure is discovered.

Disease Dice

Each color of disease die has different face values from the other colors'. Each one has a "Cross" face (value 0) and 5 other values. The average value of the faces on each die is 3 but since the values are different the standard deviations of the values on the dice are different. The values on the faces of the dice are as follows.

Value	Black	Yellow	Blue	Red
1st	0	0	0	0
2nd	3	2	1	1
3rd	3	2	2	1
4th	3	4	3	4
5th	4	5	6	6
6th	5	5	6	6
Avg	3	3	3	3
Std Dev	1.67	2	2.53	2.68

So in terms of trying to cure the diseases, the likelihood that the total of the values across all the dice you roll of a color will meet the required sum is different. Below is a table of probabilities of curing the disease with various numbers of a color of dice. The amount needed to cure a disease is normally 13, but sometimes can be 11.

	Black		Yellow		Blue		Red
# Dice	11+	13+	11+	P13+	11+	13+	11+	13+
2	0.00%	0.00%	0.00%	0.00%	11.11%	0.00%	11.11%	0.00%
3	32.87%	7.41%	34.72%	12.04%	34.26%	20.37%	40.28%	23.15%
4	70.14%	46.91%	67.67%	45.76%	60.88%	45.06%	62.73%	45.76%
5	89.51%	77.22%	86.52%	73.53%	79.90%	67.30%	78.29%	67.36%
6	96.81%	91.85%	95.03%	88.94%	90.71%	82.81%	88.70%	82.04%
7	99.12%	97.42%	98.33%	95.86%	96.05%	91.82%	94.62%	90.59%

You'll see that for certain numbers of dice and goal numbers to reach, the probability of curing the disease can be quite different. For example, with 3 dice and a goal of 13 the probabilities range from 7.41% to 23.15%. Most differences are <10%, but that can be a fairly significant difference.

You'll see that no die is universally easier or harder to find cures with. Getting 13+ is only really possible once you have 4 dice. A goal of 11 isn't very likely until you have at least 3 dice, and even then the odds are very bad. It's very hard for a single character other than maybe the generalist to amass 4 or more dice by themselves. After you have 3 dice bottled up you only have two dice left. So getting the 1 in 6 result of being able to bottle up on your dice when you only have two dice is fairly unlikely. The game allows you to trade your bottled up dice to another player if you're on the same square. This probability table tells me that that's a very important part of the game.

Advanced discussion:
In the above two tables, I ordered the dice colors by their standard deviations, lower on the left and higher on the right. One thing you might notice is that for 3 dice, the higher variance dice (aka higher standard deviation) have a higher probability of success. You'll notice that for higher numbers of dice, the colors with a lower standard deviation tend to have a higher chance of success.

When you have 3 dice, the average value of the sum is 9 (because the average for any given die is 3). Nine is insufficient for either goal so results near the average are bad. So you want a result that's far from the average, meaning you want a higher standard deviation. When you're at 5+ dice, the average result, 15, is above the goal so lower standard deviations are better.

When average is bad and you want that extreme result, you'll do better with a higher standard deviation. When the average is good and you don't need an extreme result, you'll do better with a lower standard deviation.

Player Dice, showing all faces

Epidemic Roll Change
Another place where probability plays a big role (roll?) is with epidemics. Each character die has one face which, when rolled, will advance the epidemic track. The generalist, with their seven dice instead of the normal 5, stands a much greater chance of rolling these values on their turn. To balance this, the generalist is allowed to ignore the effect of the first epidemic they roll each turn. This has a huge effect, and it makes the generalist have an overall lower change of advancing the epidemic track than other characters. Below is a table of probabilities for how far each character will advance the epidemic track on their initial roll of dice (with full dice i.e. no dice locked up from bottling up diseases).

Advancement	Normal	Generalist
0	40.19%	66.98%
1	40.19%	23.44%
2	16.08%	7.81%
3	3.22%	1.56%
4	0.32%	0.19%
5	0.01%	0.01%
6	NA	0.0004%
Avg	0.83	0.45

The other advantage of being the generalist is that when you have no epidemics on your initial roll (28% of the time w/ 7 dice, higher w/ fewer) you can freely reroll dice to try and get a better result w/ no fear of the consequences of rolling an epidemic.

Board Games!

This is actually an old picture, it's gotten much worse.

Anyone who follows me on twitter probably has noticed that I've been talking about tabletop games quite a bit lately. In the past couple of months Sarah and I have added significantly to our collection. For an idea of what I'm talking about, look to the right. It's gotten much worse since then.

We have 56 games, in total. Not all of them are in that picture, because some of them are actually behind the others. For example, you can see Munchkin there but we actually have other versions of Munchkin, they're just stashed behind Carcassonne, Ghost Stories, and Yahtzee.

There's quite a variety there, too. You see classic games like Risk and Monopoly, but there's plenty of other games too. There's the cooperative fire-fighting simulator Flash Point: Fire Rescue. There's the popular Eurogame about connecting train routes Ticket to Ride. There's the literally-only-sixteen-cards get-your-love-letter-to-the-princess simulator Love Letter (a truly excellent game). There's also the dexterity-challenging magnet-balancing game Polarity.

This might seem like a sudden shift for me but it's really a natural extension of a trend that's been going on for roughly a decade.

Forbidden Island, a cooperative game where you play as treasure hunters trying
to get four relics from an island before it sinks.

Seeking new things
I can't really say that I know what caused it. Maybe it was because of my friend the next room over my freshman year of college. Maybe it's because that was the year Katamari Damacy was released. Maybe it's because that was the year that the PSP and the DS were released, and I hadn't really been into portable gaming since the original Pokemon some long time prior. Or possibly it's because of all these things. Ever since that year, however, I've constantly been seeking new gaming experiences. I would rather play small, mediocre, yet novel games as opposed to a full-price game that's well polished yet doesn't bring much new to the table. In the past this has meant playing mobile/handheld games and download-only games, but now it's extending to tabletop games.

Tabletop games really offer a lot of things that video games don't.

Meat Space Nine
Tabletop games are all about playing with your friends right around you. You can see and talk to each other in ways that are hampered by communicating over headset or having to share real estate on a screen. This isn't to say that there aren't great video games that you can play with your friends all on the same couch, Smash Bros. and Towerfall and great examples of such, but this is what board games are all about. It is their jam.

Dungeons & Dragons: Castle Ravenloft

No need for dexterity
Tabletop games are almost always turn-based as well (Escape: The Curse of the Temple notwithstanding). Many people can't play competitive video games because of a reliance on manual dexterity, fast reaction times, having to juggle a lot of information without time to think, or they can get nauseous in the case of a first-person game. This gives tabletop games an extra level of accessibility that video games don't have.

Hackability
Most video games go to great lengths to keep you from playing them in ways that the developers don't intend. You can't make your own rules, except on a social level ("Nobody's allowed to pick Oddjob, okay!?"). You can't add and remove components. You can't do anything, usually. Tabletop games literally cannot avoid this. Don't want to play with a particular rule? GONE. Want to add your own class to the roster of characters? DO IT. Want to add a rule or more content to the game? EASY. Think something is unbalanced? CHANGE IT. They're literally powerless to stop you. This makes them great for budding game designers to experiment with how changing rules affects the gameplay or for hobbyist to make something that they love even better. If I think that Smash Bros isn't balanced well it takes a ton of effort to make it more balanced. If I think a Dungeons and Dragons class is unbalanced, that's easy to fix.

More apparent mathiness
One thing that really appeals to me in particular is, in addition to their hackability, is that their turn-based nature makes it easier to see the math behind the game and optimize your gameplay. For example, in Ticket to Ride, you get 1 point for a 1 train section, 2 for 2, 4 for 3, 7 for 4, 10 for 5 and 15 for 6. Here you can easily see that you get more points per train from doing longer routes and should try and do those if possible. This advantage becomes even more clear when you realize that by playing trains there is an opportunity cost in that any turn spent playing trains is a turn in which you aren't drawing cards. So you could spent two turns playing 3 trains each and be down 6 cards and only have 8 points or you could spend two turns, 1 playing 6 trains and 1 drawing cards, and have 15 points and only be down 4 cards.

I have by no means given up on video games. I still love and play those. Most recently I've been playing a lot of Hearthstone and Spelunky.