A Problem of Probability

I'd like to further explore a question of probability that came up in-game. In order to do so without taking the game thread way off topic, I've started this thread.


Also, I'm a big nerd who thinks probability and problem-solving are fun and interesting.


Some players have suggested that we can make informed guesses about what roles players have been assigned by examining their roles in past games. I'm asserting that history is irrelevant when calculating the probability of such things, and here's why:

You cannot calculate the probability of an event if you already know the outcome of the event. Or rather, the probability of things that have already happened is 100%.

Here's an example -

Let's say there is a series of three games of Mafia, each with 18 players total, 3 of whom are Mafia Scum. (We'll keep the numbers the same for easier math).

Now let's say we want to know the probability of Player X being selected as Scum for all three games. We do this by first finding the probability of X being scum in one game, which is 3/18, or 1/6 (Same as the odds of rolling a six on a six-sided die). To find the probability of this happening three times in a row we take 1/6 x 1/6 x 1/6, giving us a result of 1/216.

Pretty long odds, right? Thing is, they only apply when we DON'T KNOW the outcome of any of the events. Once we DO know the outcome of one or more events in our series, it changes the odds of us finding our result.

Let's use the dice analogy - the odds of rolling a six three times in a row are 1/216. We roll once, and we get a six! This result increases the odds of us rolling 3 in a row, as now we need only roll two more, the odds of which are 1/36 (1/6 x 1/6). We roll again, and amazingly get another six! Now, with two results in, we've again increased our chances of rolling three sixes in a row. Now the odds are the same as a single dice roll - 1 in 6.

So you can see that events we know the outcome of no longer have any bearing on the probability of events we do not know the outcome of.

The probability of flipping a coin to "heads" 6 times in a row is 1/64. But once you've flipped heads five times in a row, the probability of getting heads on that sixth flip is still 1/2.


The whole point of this is to illustrate that it's pointless to consider past roles when trying to decide who is scummy, because they have no statistical bearing at all on the game at hand.


(And for anyone who is wondering, I did roll a six on the third on as well, and I'm putting the 18 on Wisdom - with my racial bonuses, I'm going to have the best cleric ever.)

SW

This is a very good post. I also had similar ideas in mind but I was too lazy to organize my thoughts and put them into words.

Kudos to you, SW!!

Cleric? Cleric?

Oh man, use it and become a Paladin....I ran a Paladin for 12 years....damn he was cool, even had a Pegasus as a follower...we flew everywhere....ahhh those were the times....

Nice post

Alright alright SW, I see your point, thanks for putting me in my place. I will not use past game probabilities anymore in my post to avoid bias and confusion. But I still will use it a tiny bit in my logic, because I know how hard it is to flip a coin heads 3 times in a row. It is possible to flip 20 heads in a row, but I have never seen it done I stress a TINY bit because it is only 1 small factor of many factors I use to analyze the data.

On one of Derrin Browns shows here last year, he flipped I think it was 10 heads in a row. He explained about the proability of doing so. It took them something like 8 or 9 hours constant filming and flipping before he got the 10 in a row.

It's probably on a torrent somewhere, I think it was called The System, and had to do with race horses and betting etc.

@ thefiend - wasn't trying to put you in your place so much, just to demonstrate why that particular piece of figuring (unfortunately) doesn't always work so well.

@stimp - my interest in probabilities is also mainly derived from gambling...


@talhun - I was just goofin with the cleric, but yeah paladins were badass, a bit too 'holy' for my tastes, i always had ranger/druid/monk tendencies.

When considering balance while designing a game, I began to think about the odds of a doctor protecting the mafia target. I began with a basic setup of 1 mafia kill, 1 doctor who cannot self protect, and 0 role blockers. Here's the equation that resulted:

(T-1)/(T*(P-1))

The number of town minus one (players that could be targeted by both), divided by the number of town (mafia targets) multiplied by the number of players minus one (doctor targets).

Thus a basic 9 player setup of 2 mafia goons, 1 doctor, and the rest town (cops don't factor into this probability) with a town lynch going into the first night . . . the odds that the doctor will protect the player who is targeted by the mafia is:

(T-1)/(T*(P-1))
(6-1)/(6*(8-1))
5/(6*7)
5/42
11.9%

I am planning to extend this equation to include factors such as role blockers and multiple night kills. Let me know if you find this interesting, useful, a waste of time, whatever.

Moving on to the next basic setup: 1 Godfather (role blocker), 1 mafia kill, and 1 doctor who cannot self protect. The resulting expression is only slightly expanded:

((T-1)*(T-2)) / (T*(T-1)*(P-1))

The number of town minus one (players that could be targeted by both the doctor's protection and the mafia's kill) multiplied by the number of town minus two (a factor of remaining two non-doctor players so that the doctor is neither killed nor role blocked), divided by the number of town (mafia night kill targets) multiplied by the number of town minus one (mafia role blocking targets) multiplied by the number of players minus one (doctor targets).

This can, of course be reduced to the following expression:

(T-2) / (T*(P-1))

If night actions are applied in the order of role blocking, protection, then kills and assuming a town lynch going into the first night with the basic 9 player setup . . . the odds that the doctor will protect the player who is targeted by the mafia, and is neither role blocked nor killed is:

(T-2) / (T*(P-1))
(6-2) / (6*(8-1))
4 / (6*7)
4 / 42
9.5%

This is a lower percentage as compared to our no role blocker setup. If it is a closed or semi-open setup, that decrease is justified by it leading to the knowledge of a role blocking player.