Pandemic: different types of randomness

who: J1
where: my home office
when: after lunch

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We've been playing pandemic a lot recently.

For this post, you don't really need to know the game, just that there are a deck of playing cards from which each player draws on their turn. Usually, these are resources that are needed to win the game, but occasionally there are epidemic cards that really stink. Oh, also know that everyone is playing together, cooperatively, against the game itself.

J1 noticed something very interesting by varying the set-up:

• Normal set-up: split the playing cards into equal piles and put one epidemic card into each pile (number of piles is the number of epidemic cards you are using). Shuffle those piles and then stack them on top of each other.
• Modified set-up: put the epidemic cards you are using in the deck and then shuffle the whole deck.

Focusing just on the epidemic cards, these result in very different distributions. Here are some challenging questions (assume there are 5 epidemic cards and 50 non-epidemic playing cards):

1. For both distributions, what is the probability that the n-th card in the deck is an epidemic card?
2. For both distributions, what is the probability that two cards in a row are epidemic cards?
3. For both distributions, what is the probability that three cards in a row are epidemic cards?
4. For both distributions, what is the probability that all epidemic cards are in a row?
5. What have we learned about how these distributions compare?

Ok, these are hard questions, particularly if you work through them in order 1 - 5. Without giving away the answers, I think it is striking that:

1. the answer is the same for both distributions, but what follows is very different!
2. left for reader
3. much higher for modified set-up
4. actually not too hard to calculate for each distribution; again, much higher for modified set-up
5. very different for potential impact on game play

Conditional probabilities
I'm open to other suggestions, but currently think that the key way to see the difference in these two ways of shuffling is to focus on conditional probabilities. For method 1, say p1(n|i) is the probability that the nth card will be an epidemic card, given that there have been i epidemic cards already drawn up to that point and similar for p2(n|i). For a fixed i, even the domains of definition of p1 and p2 aren't the same!

What happened when we played?
There's one key fact you need to know to understand our actual game: J1 hasn't learned how to really shuffle yet. That's right, all the epidemic cards were in one cluster!

J1 asked me to include some other information about our game play:

• J1 really likes to be the medic. At the start of the game, it is the most helpful piece
• Dispatcher plus medic is a very powerful combination. The dispatcher becomes even more powerful than the medic once cures have been found.
• We don't really follow the hand-limit rule.  This feels like an unnatural restriction that doesn't fit with the story of the game.