The hardest part of computing the probability is determining the best strategy for the players. I have used a script to simulate games where players use a few simple tactics, and estimated the probability based on these strategies. Given that the strategies I implemented are not the optimal strategy, these figures provide an upper bound on the probability of a first turn loss.
Most of this answer assumes that the players are playing on "Normal" difficulty, that is with five epidemic cards. To find equivalent probabilities for other difficulties, see the section below entitled "The player deck".
To summarise the results, here is a table showing my estimate of the probability of a first turn loss for different strategies.
Strategy | Probability of first turn loss at Normal difficulty
Do Nothing | 0.0018
Drive/Treat | 0.0013
Drive/Fly/Treat | 0.0011-0.0012
The probability of losing in the first turn, given that the players do their best to prevent it, is therefore less than 1/900.
The Player Deck
To set up the player deck, some cards are dealt to the players and then epidemic cards are distributed throughout the deck. The number of cards dealt to the players, d, varies with the number of players. We either have d = 8 in 2 or 4 player games, or d = 9 in 3 player games. The remaining (53-d) cards are split into e roughly equal piles, and an epidemic is shuffled into each, before they are restacked (with any larger piles on top).
We observe that to lose in the first turn an epidemic must be drawn. The probability of drawing an epidemic as one of the top two cards of the deck is:
P(Epidemic drawn) = 2 / (ceil((53-d)/e)+1)
where ceil is the ceiling function. This results in the following table:
d | e | P(Epidemic drawn)
8 | 4 | 2/13 = 0.154
8 | 5 | 2/10 = 0.200
8 | 6 | 2/9 = 0.222
9 | 4 | 2/12 = 0.167
9 | 5 | 2/10 = 0.200
9 | 6 | 2/9 = 0.222
During the rest of the discussion we assume e is 5 ("Normal" difficulty), but the above table can be used to compute equivalent probabilities for other values of e and d.
The "Do Nothing" Strategy
By far the simplest strategy is for the players to do nothing. This is also a strategy that maximises the chance of losing in the first turn.
For the calculation I employed a python script to simulate a million games, assuming that an epidemic card is drawn. It picks nine cities to infect for the set up, and then one more to infect for the epidemic. It repicks two cities to re-infect, and counts the outbreaks and cubes used.
This gave the probability of losing, given that an epidemic is drawn is about 0.009245.
Combining this with the probability of drawing an epidemic card we get:
P(Lose first turn on Normal | Players do nothing) ~= 0.001849
The "Drive/Treat" Strategy
As a simple improvement on doing nothing, I assumed the first player would ignore their cards, and drive to the nearest city with three cubes, removing as many as possible. If they couldn't remove any from three cube cities, they would try for the nearest two cube city, and finally they would try for the nearest one cube city.
This gave the following probability:
P(Lose first turn on Normal | Using "Drive/Treat" Strategy) ~= 0.001343
The "Drive/Fly/Treat" Strategy
Finally, as an improvement on the "Drive/Treat" strategy I considered trying to fly to a city infected with three cubes. If anyone has the "Airlift" or "Government Grant" special event cards, then they can be used to get the first player to any city in 0 or 1 actions respectively, meaning that three cubes can be treated. Similarly if any other player has the Atlanta card it can be passed, and the first player can charter a flight, leaving two actions to remove cubes.
I also considered the case where the player has the card of a 3-cube city, or Atlanta, and they use that to fly and remove three cubes from a city.
There are further variations on this that I did not consider in this tactic, such as moving to Washington, chartering a flight and treating two cubes.
Using this strategy gave the following probabilities:
P(Lose first turn on Normal | 4 players using "Drive/Fly/Treat" Strategy) ~= 0.001208
P(Lose first turn on Normal | 3 players using "Drive/Fly/Treat" Strategy) ~= 0.001145
P(Lose first turn on Normal | 2 players using "Drive/Fly/Treat" Strategy) ~= 0.001167
I don't think my simulation is evidence enough to say that the deviation between these three results is significant, but I think they're good enough to give a first-turn-lose probability between 0.0011 and 0.0012.
If you have played the game then I'm sure you can think of situations in which these strategies are not optimal. Here are a few things that could be included, that I think would make a significant improvement to the strategies:
- Treat more than one city if possible.
- Target infected cities that are next to each other in preference to isolated ones.
- Use "One Quiet Night" or "Forecast" to prevent the first turn loss (this is a big one - it's possible about one game in three).
- Use "Resilient Population" to lower the chance of a first turn loss.
- Take the researcher into account for passing cards.