Oh my god. I am actually using my degree.
Before I begin, I just wanted everyone to know that I am going to try my best to not spoil any of the game other than what you might have seen in the Prologue month.
Understanding Probability in Pandemic Legacy: Season 2 is a multi-part series. Links to the other parts can be found here:
Understanding Probability in Pandemic Legacy: Season 2 – Part 1
Probability of Incidents During Set-Up
Understanding Probability in Pandemic Legacy: Season 2 – Part 2
Probability of Incidents During Set-Up and How-To Use the Front 9 Calculator
Understanding Probability in Pandemic Legacy: Season 2 – Part 3
Strategies to Survive Between Set-Up and First Epidemic
Understanding Probability in Pandemic Legacy: Season 2
Game play of the Prologue month can easily be found on YouTube and the month is often used to introduce the game play, mechanics, and work flow to new players. The prologue has no permanent effects on the game and can be played an unlimited number of times before starting the January month.
For those of you who are in a current game of Pandemic Legacy: Season 2, you might have noticed that the game is quite a bit harder than Season 1. You aren’t alone. It seems like everyone is struggling to meet their monthly objectives. Here are some tips.
Optimize Your Chances of Winning
- Review the contents of the Infection Deck. Count the number of cards and see how the city infection cards are distributed. At the beginning of the game, there are 27 cards in the infection deck. 9 cities with 3 card each. This will change as the game progresses. As you are playing the game, keep track of which cards came up. This will help you determine the likelihood of future cards.After each epidemic, before shuffling the infection deck discard and putting it on top of the infection deck, review the played cards and take notes. Again, this will help you prepare for the upcoming infections.
- Review the contents of the player deck. Most importantly, get an idea of how often an Epidemic will come up. If the player deck has 60 cards, and there are only 5 Epidemics in the game, you should expect an Epidemic in each 12 card cycle.
- Take Notes. Tally the number of players cards that you have gone through. Tally the number of cities that have come up. Tally how many turns/player cards since the last epidemic, or which epidemic cycle you are currently on.
Solving for the Probability of Event (blank)
Now onto the probability. Why is probability so important? A core mechanic of Season 2 is keeping enough supply (gray) cubes in each city on the board to adequately protect the city from incidents, or infection of a city when there are no stockpile cubes left. When a city gets infected, a supply cube is removed. When a city is infected and has no supply cubes left, a disease cube is placed in the city and the incident marker is moved forward. When the game reaches the 8th incident for the game, Pandemic has won and that attempt for that month is over.
How do we determine how many stockpile cubes are adequate? How do we distribute stockpile cubes in such a way that we minimize the number of incidents occurring during the initial infection?
That’s really a great question. To best answer that, we need to make some assumptions as well as simplify the scenario a little bit.
During the set-up of the game, the players have a limited number of supply cubes to distribute among the cities connected to the grid. For the prologue month, there’s 9 cities and 36 stockpile cubes that need to be distributed. The infection deck has 27 cards. 3 cards for each city connected to the grid. After the cubes are distributed, set-up does an initial infection by drawing 9 cards from the infection deck and infecting the 9 revealed cities.
Solution (Kind of.)
To answer the initial question, let’s simplify even more. Let’s only concentrate on a single city, New York. Of the 27 cards in the infection deck, 3 cards are New York. During the set-up infection (9 infection cards), what’s the likelihood that 0, 1, 2, or 3 New York cards show up? Does it make sense to place 0, 1, 2, or 3 cubes on New York to prevent incidents?
Defining some variables:
Deck size = N = 27
Total number of cards drawn = R = 9
Number of New York cards = n = 3
Number of New York Cards drawn = r = [0, 1, 2, or 3]
If we were only drawing one card, probability would be really easy to determine. P(X = Draw New York) = 3/27.
However, since we are drawing 9 infection cards, we have to factor in that New York cards can fall into any slots. We could draw a New York with the first card, or the last card, or any card in between. In order to account for this, we will use combinations as don’t care about order.
Some other assumptions: Though the remaining 24 cards in the Infection Deck are not all the same, for this problem, we’ll just lump them together and consider them “Non-New York.”
More information on Combinations can be found here under “Combinations, Ho!”
Combinations = nCr = C(n, r)
Solve in Google Sheets using this function: =combin(n, r)
P(X = r) =
C(n, r) * C(N-n, R-r)
In layman’s terms,
C(n, r) is the number of ways to draw r New York card from n New York cards.
C(N, R) is the number of ways to draw R (R=9) cards, from N (N=27).
C(N-n, R-r) refers to the number of ways to draw R-r (Non-New York) from N-n (Non-New York)
P(X = Draw 1 New York)
P(X = Draw 1 New York) =
[Combination(of drawing 1 New York from 3 New York cards) x Combination(of drawing 8 Non New York cards from 24 Non-New York Cards)] / [Combination(of drawing 9 cards from 27)]
It should look something like this:
C(3,1) * C(24, 8)
P(X = Draw 1 New York) = 0.4708
P(X = Draw 2 New York)
Let’s try for P(X = Draw 2 New York).
C(3,2) * C(24, 7)
As you can see, only the numerator changes. C(3,2) refers to drawing 2 New York’s from a total of 3. While C(24,7) refers to the number of ways to draw the 7 remaining cards from the Non New York subset.
P(X = Draw 2 New York) = 0.2215
P(X = Draw 3 New York)
C(3,3) * C(24, 6)
P(X = Draw 3 New York) = 0.0287
P(X = Draw 0 New York)
Just for due diligence, let’s also do P(does not draw New York).
C(3,0) * C(24, 9)
P(X = 0 New York) = 0.2790
P(X = 0) = 0.2790
P(X = 1) = 0.4708
P(X = 2) = 0.2215
P(X = 3) = 0.0287
Let’s interpret the results.
P(X = 0) = 0.2790 – Probability that New York gets exactly 0 infections
P(X = 1) = 0.4708 – Probability that New York gets exactly 1 infection
P(X = 2) = 0.2215 – Probability that New York gets exactly 2 infections
P(X = 3) = 0.0287 – Probability that New York gets exactly 3 infections
Ex. I put only supply cubes on New York. What’s the risk or likelihood that New York will have an incident during the initial infection set-up? (What’s the probability that New York will be infected 3 times during Set-Up?)
P(X > 2) = 1 – P(X > 2) = P(X = 3) = 0.0287
Ex. I put one supply cube in New York. What’s the risk or likelihood that New York will have an incident during the initial infection set-up? (What’s the probability that New York will be infected 2 or more times?)
P(X > 1) = P(X = 2, 3) = P(X = 2) + P(X = 3) = 0.2215 + 0.0287 = 0.2502
Ex. There are only two New York Infection cards in the deck. (2 New York cards and 25 Non-New York cards) What are the probabilities of P(X = 0), P(X = 1), and P(X = 2)?
Deck size = N = 27
Total number of cards drawn = R = 9
Number of New York cards = n = 2
Number of New York Cards drawn = r = [0, 1, or 2]
The set-up of the formula is the same except that we’ll be using the new numbers above to substitute them into the combinations formula.
For P(X = 0), the formula will look like this:
C(2,0) * C(25, 9)
P(X = 0) = 0.4359
Furthermore: P(X = 1) = 0.4615 and P(X = 2) = 0.1026.
Set-Up Infection Calculator in Google Docs
I’ve set up this cool little table/calculator to determine the probability that cities will become infected with 0, 1, 2, or 3 infection cubes during set-up. Just update your Infection Deck size in the green box.
What’s the best way to distribute supply cubes during set-up? Using the Set-Up Infection Calculator and the combination theories above, the best guidelines are to distribute supply cubes in this order. Continue to the next point if you still have stockpile cubes still available. Proof of this strategy can be found in Part 2.
1. Place one supply cube in each city that has 3 cards in the infection deck.
2. Place one supply cube in each city that has 2 cards in the infection deck.
3. Place one supply cube in each city that has 1 card in the infection deck.
4. Place a supply stockpile cube in each city that has 3 cards in the infection deck.
5. Place a supply stockpile cube in each city that has 2 cards in the infection deck.