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Subject: 2 player game: starting with a cure on hand rss

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(ɹnʎʞ)
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Hallo everyone,

when playing Pandemic with 2 players, every player starts out with 4 cards.


Since the scientist only requires 4 cards to research a cure, it it theoretically possible that this player could start out the game with already a cure on his/her hand.

I'm curious. Has this ever happened to anyone ?

Poll
Did you ever start a 2-player game with the scientist already having a 4-card cure on his/her hand ?
Yes
No
      128 answers
Poll created by Kyur


Since all players also start at a research station, it is even possible to cure a disease as the very first action in a game -- and even further, theoretically also possible to eridacte a disease completely right from the beginning.
 
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Michael Tyree
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I have once, but I know the culprit was a bad shuffle of a fresh deck. Right from the package, the colors are all in a run. The gent shuffling his new cards was excited to play, didn't do more than a few passes and we had long stretches of the same colors. Whoops, live and learn

Now, for well mixed and shuffled decks I have had games where I could have a first turn cure, but not just on the scientist's hand. Mixed over folks hands at least once was a blue cure by trading ATL and another was the powerful scientist/researcher role combo.
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Annemarie Post
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Last game we played 2-player with the Scientist, she did get 3 cards of the same color in her first hand, and the other player had the 4th. That was the easiest first cure we ever had.
 
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Ray
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Is it possible? Yes, but I'd probably re-shuffle and deal new cards. I like the game to be somewhat challenging and would be disappointed if one cure was already available from start. I also would be afraid it was because of poor shuffling, though I'd be interested to see someone do the math on the chance of having 4 cards dealt to you of all the same color when factoring in epidemics and action cards.
 
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Cory Kneeland
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Not only did we have this happen, we eradicated the disease as well due to unique initial disease set up. It was cool. Game was still a blast, perhaps a tad easy, but fun.
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George I.
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The probability to start with a cure in hand is extremely low? Can you guess the magnitude? It's less than 1%; it's as low as... 0.73%.

Rationale: there are 52 cards in the deck: 48 cities and 4 events (not 5, I'm using the OTB rules). This makes us C(52,4)=270,725 different 4-card hands.

We have 12 cards of each color and C(12,4)=495 different hands of 4 same-colored hands. We can multiply by 4, since we have 4 colors, as all these hands are different (different colors), i.e., we are not adding twice any same hand: 4*C(12,4)=1,980 different hands of 4-colored cards.

Thus, the probability is 1,980/270,725 or 0.73%.

This can be adjusted to account for two players drawing cards or the probability of being dealt with the Scientist, as well, but you get the point.
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Annemarie Post
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Picon wrote:

Rationale: there are 52 cards in the deck: 48 cities and 4 events (not 5, I'm using the OTB rules). This makes us C(52,4)=270,725 different 4-card hands.


Don't forget the epidemic cards, but thanks for the calculation
 
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Désirée Greverud
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Picon wrote:
The probability to start with a cure in hand is extremely low? Can you guess the magnitude? It's less than 1%; it's as low as... 0.73%.

Rationale: there are 52 cards in the deck: 48 cities and 4 events (not 5, I'm using the OTB rules). This makes us C(52,4)=270,725 different 4-card hands.

We have 12 cards of each color and C(12,4)=495 different hands of 4 same-colored hands. We can multiply by 4, since we have 4 colors, as all these hands are different (different colors), i.e., we are not adding twice any same hand: 4*C(12,4)=1,980 different hands of 4-colored cards.

Thus, the probability is 1,980/270,725 or 0.73%.

This can be adjusted to account for two players drawing cards or the probability of being dealt with the Scientist, as well, but you get the point.

so once every ~137 2-player games will one of the players start with 4 of the same color. in base game, that person has a 1 in 7 chance of having the scientist, so that's a 1 in ~959 games chance of starting with a cure. This becomes even rarer as you add in more roles.
 
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Jim Parkin
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Yep, this has happened to me before.

We still lost...
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Ethan Furman
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Anny48 wrote:
Picon wrote:

Rationale: there are 52 cards in the deck: 48 cities and 4 events (not 5, I'm using the OTB rules). This makes us C(52,4)=270,725 different 4-card hands.


Don't forget the epidemic cards, but thanks for the calculation


The epidemic cards are not in the deck when the players' hands are dealt, so they don't factor in.
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Annemarie Post
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stoneleaf wrote:
Anny48 wrote:
Picon wrote:

Rationale: there are 52 cards in the deck: 48 cities and 4 events (not 5, I'm using the OTB rules). This makes us C(52,4)=270,725 different 4-card hands.


Don't forget the epidemic cards, but thanks for the calculation


The epidemic cards are not in the deck when the players' hands are dealt, so they don't factor in.


Good point, thanks
 
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Simon Kamber
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DragonsDream wrote:
so once every ~137 2-player games will one of the players start with 4 of the same color. in base game, that person has a 1 in 7 chance of having the scientist, so that's a 1 in ~959 games chance of starting with a cure. This becomes even rarer as you add in more roles.


Not quite. There's two players in the game, so you'll have one of the players start with four-of-a-kind in one out of ~69 games. That player will then be the scientist in one out of 483 games.
 
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Andy Burgess
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Randomness is clumpy.
 
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Matthew Burgess
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Dulkal wrote:
That player will then be the scientist in one out of 483 games.

I don't understand this comment. There is a much higher chance of drawing the Scientist than once every 483 games...
 
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Byron S
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mattpburgess wrote:
Dulkal wrote:
That player will then be the scientist in one out of 483 games.

I don't understand this comment. There is a much higher chance of drawing the Scientist than once every 483 games...

This is the chance that the person both draws the Scientist AND has 4 cards of a single color in the initial deal.
 
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David Goldfarb
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Anny48 wrote:
Last game we played 2-player with the Scientist, she did get 3 cards of the same color in her first hand, and the other player had the 4th. That was the easiest first cure we ever had.

Unless the other player was the Researcher, you still had to both get to that other city, and then get from there back to Atlanta. (Or get a research station built and get to that.) Not so very easy...if you're playing by the rules. If the other player was the Researcher, then yeah.
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James Patterson
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Annowme wrote:
Yep, this has happened to me before.

We still lost...


My son and I are getting ready to launch Legacy, so we've played a couple games with the vanilla rules to get a feel for it, as recommended for those who haven't played Pandemic before. We weren't dealt a cure, but he started with a lot of black and had the cure after the first couple of turns. Odd thing was, no black cities were revealed during the initial infections, and none had come up yet, so the disease was eradicated as well. We thought, sweet! Until no black city cards came up for a very long time, so we didn't get the free infection draws we were hoping for, and neither of us could get a set of 5 for quite a long time. We ended up losing with only 2 diseases cured as the deck ran out.

Admittedly, we're still figuring things out, but I wouldn't assume an early cure will guarantee an easy game.
 
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(ɹnʎʞ)
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Interesting.

Apparently it happened more often than I expected (based on the current poll result of 21.6% Yes and 78.4% No).

I played this game a lot and it never happened so far.
 
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George I.
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Picon wrote:
The probability to start with a cure in hand is extremely low? Can you guess the magnitude? It's less than 1%; it's as low as... 0.73%.

Rationale: there are 52 cards in the deck: 48 cities and 4 events (not 5, I'm using the OTB rules). This makes us C(52,4)=270,725 different 4-card hands.

We have 12 cards of each color and C(12,4)=495 different hands of 4 same-colored hands. We can multiply by 4, since we have 4 colors, as all these hands are different (different colors), i.e., we are not adding twice any same hand: 4*C(12,4)=1,980 different hands of 4-colored cards.

Thus, the probability is 1,980/270,725 or 0.73%.

This can be adjusted to account for two players drawing cards or the probability of being dealt with the Scientist, as well, but you get the point.

And an update on that: let's adjust the probability to account for TWO players now. What's the probability of at least one player drawing a 4-colored hand?

Let's call this probability P(A or B). It's calculated as P(A or B) = P(A) + P(B) - P(A and B). Intuitively, you can just add the two probabilities together, but you have to subtract the probability of BOTH players drawing great hands, as we've added these hands twice; i.e., in P(A)+P(B) we have added twice the hands where both players will have a 4-colored hand.

So what's the probability of both players having a 4-colored hand?
¶1) Player A has P(A)=4*C(12,4)/C(52,4) chances to get a 4-colored hand (e.g., red).
¶2) Player B can either get:
¶2.1) a 4-colored hand of a different color (e.g., blue/yellow/black), or...
¶2.2) a 4-colored hand of the same color (e.g., red), but out of the remaining 12-4=8 cards of that color.

In case ¶2.1, there are 3 colors remaining, with 12 cards each. Thus, there are 3*C(12,4) 4-colored hands, of the rest of the colors. But, since we have now 48 cards in the deck, as player A already has drawn 4 cards. Thus, there are C(48,4) hands remaining, therefore, the probability here is 3*C(12,4)/C(48,4)=3*1,980/194,580 (about 3.05%).

In the latter case ¶2.2, there are 12-4=8 cards remaining of the same color as player A has. There are therefore C(8,4) different hands and in total C(8,4)/C(48,4)=70/194,580 (about 0.36‰; note, that's less than 1 in a thousand games).

To have either ¶2.1 or ¶2.2: since they are mutually exclusive, we just add these together. P(¶2.1 or ¶2.2) = P(¶2.1) + P(¶2.2) = 3*C(12,4)/C(48,4) + C(8,4)/C(48,4) = 311/38,916. This is just the second player drawing a 4-colored hand out of the remaining cards, but we have not accounted for the first player to draw such a hand, yet.

P(A and B), the probability to have both players drawing a 4-colored hand, is P(¶1 and ¶2) = P(¶1 and (¶2.1 or ¶2.2)), i.e., the probability of one player drawing such a hand and the other player as well, from what's left of the deck. Since these two probabilities are mutually exclusive (as we have removed the cards that player A has drawn), it's:
P(¶1 and (¶2.1 or ¶2.2)) = P(¶1) * P(¶2.1 or ¶2.2) = P(A) * P(¶2.1 or ¶2.2)

That's about 5.84e-5, i.e., both players drawing a 4-colored hand will occur once every 20,000 games.

What if either one draws a good hand? That's:
P(A or B) = P(A) + P(B) - P(A and B)
= P(A) + P(A) - P(A) * P(¶2.1 or ¶2.2)
= P(A) * (2 - P(¶2.1 or ¶2.2))
= 4*C(12,4)/C(52,4) * [2 - (3*C(12,4) + C(8,4))/C(48,4)]
= 852,731/58,530,745
= 1.47%


Thus, the probability of either player to draw a 4-colored hand is 1.47%, about 3 out of 200 games.

Note that this is very close to just P(A)+P(B), i.e., without subtracting the cases where both players have a 4-colored hand, as this is virtually impossible to happen!
 
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Simon Kamber
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Picon wrote:

Thus, the probability of either player to draw a 4-colored hand is 1.47%, about 3 out of 200 games.


But it's not enough for either player to draw a 4-of-a-kind hand. That player also needs to be a scientist.
 
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George I.
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Dulkal wrote:
But it's not enough for either player to draw a 4-of-a-kind hand. That player also needs to be a scientist.
It's not enough to find a cure out of a 4-of-a-kind hand. If that's your point, then you are right. Nevertheless, I am not calculating the probability of curing a disease; I was explicit in that I am calculating the probability of drawing such a hand. In about 3 out 200 games at least one of the two players will draw such a hand; doesn't mean they will be able to cure it as well.

To cure, this has to be adjusted to the following scenarios:
- The player that has a 4-of-a-kind hand is the Scientist, or...
- One player has 4 blue cards and the other player has Atlanta, or...
- The player with the 4-of-a-kind hand is the Epidemiologist and the second player has a 5th card of that color, or...
- The other player is the Researcher and both of them have 5 cards of that color, in any combination.

If you really want to stretch things out and demand a turn 1-2 cure, you can restrict in the last scenario that:
a) The non-Researcher has at least 2 cards of that color in hand, (or at least 1, if they are the Generalist), so that they can share and cure on the same turn, or...
b) The Researcher goes first.
 
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Simon Kamber
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Picon wrote:
It's not enough to find a cure out of a 4-of-a-kind hand. If that's your point, then you are right.


Well, that was the question that started the thread:
""Did you ever start a 2-player game with the scientist already having a 4-card cure on his/her hand ?""

If we are just looking for first-turn cures, the odds get a great deal better.
 
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