Conditional Probability and Slaying Werewolves

[Ater Nox]

When your town is besieged by werewolves, and you're worried you might be their next lunch, there's only one thing you can count on: Maths!

In this class, we'll learn about calculating conditional probabilities, and manipulating them to determine who needs a date with the hangman's noose.

The goal in the class is to provide the students with a new way of thinking in Werewolf games, and teach them about probability. Students will require a minimal background in mathematics, with the most advanced maths being algebraic substitution and rearrangement of simple equations.

There will be six lessons, each with a short assignment.

1. Introduction to probability: 10000islands.proboards.com/post/1494957/thread
2. Conditional probability
3. Using Bayes theorem
4. Advanced applications of Bayes theorem
5. Quantifying uncertainty
6. Final review and comments

Sign ups will close in a week or two, I will give advanced notice of the start date.

[Ater Nox]

Tagging some recent WW players who may be interested:

Dominion of Compassion
Woonsocket
Sauron
Alkasia
Flag Flayers
Control
Louisistan
Hakketomat
Nate NBNS
Free Guns For All
Tanzoria
SweetHaven
Hahiha / USSR
Sorya
RairAria
Boltor
Elite
The Kerguelen Archipelago
Palman
Ayunli
Matilda Love

[Tanzoria]

Hh, whatever, I'll do it. Just don't expect much

[Dominion of Compassion]

Sure!

[Woonsocket]

Quantum uncertainty? I'm in. Of course, since you know my position you can't know what direction I'm going with this.

[Sorya]

This sounds like fun. Count me in!

[Louisistan]

Probability of me participating? 100%

[Hahiha / USSR]

I'm in!

[Control]

Sounds interesting. I'm in.

[Alkasia]

Oh boy, let's hope my advanced math courses throughout most of my schooling have prepared me for this.

I'm in.

[Sauron]

I'm in... but this just looks like the stuff of nightmares.

I'm not even sure I have a basic level of maths!

[Hahiha / USSR]

Apr 24, 2017 15:00:27 GMT -5 Sauron said:

I'm in... but this just looks like the stuff of nightmares.

I'm not even sure I have a basic level of maths!

Who needs maths when you have eight (eleven? twenty?) human lords under your influence?

[Flag Flayers]

I'm in for it

[Palman]

MWAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHA! I'm SO in!

Will this course cover quantum werewolves variant (that I've most definitely haven't considered running on here just to %^*# with people)?

[Hahiha / USSR]

That is so mean, Palman.

[Palman]

Apr 24, 2017 16:44:14 GMT -5 Hahiha / USSR said:

That is so mean, Palman.

Oh it's delightful! Everyone chooses to attack someone at night, and who the wolves kill is chosen at random from everyone's targets. The victim is declared a civilian. Both of these change the odds that any given person is a wolf. Everyone votes to lynch someone. The lynch victim is a wolf or citizen at random, weighted based on the odds they were either. Then everyone declares a target again the next night phase. It's viscous and trippy, but mathematically beautiful. You have to keep switching back and forth between thinking like a wolf and thinking like a villager.

[Elite]

Sure, it'll be a nice break from my regression and calculus assignments.

[Free Guns For All]

If I'm going to be hosting a future game, I might as well get back into the swing of playing these. Sign me up.

[SweetHaven]

Ohhh this is going to be good.

Yup I am in

[now I really need to finish up  planning the game I will be running in a different forum ]

[Ater Nox]

We'll start Tuesday next week my time, or Monday night for you Americans. Anyone interested in joining after that will be welcome to jump in late.

[Ater Nox]

Lesson 1. Introduction to probability


Welcome everyone to the first lesson of Conditional Probability and Slaying Werewolves. In this lesson, we'll be covering the basics of probability, notation and the terminology we'll need to know.

If you have any questions, please post them in the thread. The next class will be held in a week.

Events

An 'event' is the word we are using to describe the thing we want to know the probability of. We want our events to be able to be true or false. 'Aersoldorf is a werewolf' is an event for example. We can define as many events as we want.

Each event should just be one concept or idea. If your event contains the word 'and' or 'or', or something along those lines, you should split them into separate events. For example, 'Aersoldorf is a werewolf who drives a cool sports car and enjoys monthly holidays in the Bahamas' is a poor definition of an event, but the three events 'Aersoldorf is a werewolf', 'Aersoldorf drives a cool sports car' and 'Aersoldorf enjoys monthly holidays in the Bahamas' are good.

To avoid having to write the statements out again and again, we can just denote them by a capital letter:

Let A be the probability that Aersoldorf is a werewolf.
Let B be the probability that Aersoldorf drives a cool sports car.
Let C be the probability that Aersoldorf enjoys monthly holidays in the Bahamas.

Probability

The probability of an event is the chance of it occurring. One way it can be thought of is the number of ways the event can occur divided by the number of total possibilities.

For example, if 10 people were playing a game of Werewolf, and 2 of them were randomly selected to be Werewolves, then the probability of Aersoldorf being a werewolf is 2/10. If you wanted, you could then simplify that to 1/5 or 20%.

We would write this mathematically like so:

P(A) = 2/10

Complement

The complement of an event is the probability the event does not occur. To calculate this number, we just need to subtract the probability of the event from one.

Continueing the example from before, the probability of Aersoldorf being a werewolf is 2/10 so the probability of Aersoldorf not being a werewolf is 1 - 2/10, or 8/10.

We would write this mathematically like so:

P(Not(A)) = 1 - 2/10
              = 8/10

Independence

Two events are independent if one event actually happening does not effect the probability of the second event. For example, the probability of Aerso being a werewolf is independent of the probability that I drive a red car. Regardless of whether or not Aerso is a wolf, the probability that my car is red is going to be the same. Likewise, the colour of my car not effect Aerso's probability of being a werewolf.

If two events are independent, then the probability of both events happening is just the probability of each multiplied together.

Let A be the probability that Aersoldorf is a werewolf.
Let B be the probability that Ater Nox has a red car.

P(A and B) = P(A) * P(B)

If the events are independent, then their complements will be as well.

P(Not(A) and B) = P(Not(A)) * P(B)
                      = (1 - P(A)) * P(B)

P(A and Not(B)) = P(A) * P(Not(B))
                       = P(A) * (1 - P(B))

P(Not(A) and Not(B)) = P(Not(A)) * P(Not(B))
                             = (1 - P(A)) * (1 - P(B))

We can also look at the complement of two events occurring together to get the probability of 

P(Not(A and B)) = 1 - P(A and B)

                       = 1- (P(A) * P(B))

So, are P(Not(A) and Not(B)) and P(Not(A and B))  the same? No, there is a slight difference between them. 

P(Not(A) and Not(B)) is the probability that A does not occur and B does not occur.

P(Not(A and B)) is the probability that A and B do not happen together, ie either one or neither occur.

Assumptions

We don't know everything. Sometimes we'll need to make assumptions to simplify things for ourselves, and that is ok as long as we mention the assumptions we've made.


Example

There is a werewolf game with 14 players and 3 wolves. In the first phase, someone will be elected mayor. We will assume that being a wolf and being elected mayor are independent, and each player has an equal chance of being elected mayor or being a wolf.

What is the probability that Aersoldorf is elected mayor, and is not a wolf?

First, let's define our events.

Let A be the probability that Aerso is a wolf.
Let M be the probability that Aerso is elected mayor.

From the question, we know:
P(A) = 3/14
P(M) = 1/14

We want to find the probability that Aersoldorf is elected mayor, and is not a wolf, which is:

P(M and Not(A))

Because these events have been assumed to be independent, we can multiple them together:

P(M and Not(A)) = P(M) * P(Not(A))
P(M and Not(A)) = P(M) * (1 - P(A))

Substituting our numbers in, we get:

P(M and Not(A)) = 1/14 * (1 - 3/14)

Putting that into a calculator or www.wolframalpha.com gives us an answer of 

P(M and Not(A)) = 11/196
                       = 0.0561 or 5.61%


Homework

Please post your answers to the homework questions in the thread, including any working you do. Everyone may participate in the discussion between now and the next class. 

Question 1.
A werewolf game has 22 players including 5 wolves, 2 defenders and 1 seer. What is your probability of being assigned a wolf role? What is your probability of being assigned a defender role? What is your probability of being assigned the seer role?

Question 2.
Give an example of two independent events, and two dependent events.

Question 3.
There is a werewolf game with 15 players and 1 seer. In the first phase, one player will be lynched. We will assume that being the seer and being lynched are independent, and each player has an equal chance of being the seer or lynched. What is the probability that Aersoldorf is not the seer, but is lynched on the first day?


Discussion 1.
In the example, Aersoldorf had a 5% of being elected Mayor and not being a Wolf. If he is elected Mayor, should we trust him or lynch him? What about if he is not elected Mayor?

[Ater Nox]

Tanzoria, Dominion of Compassion, Woonsocket, Sorya, Louisistan, Hahiha / USSR, Control, Alkasia, Sauron, Flag Flayers, Palman, Elite, Free Guns For All, SweetHaven

[Control]

SPOILER: Click to show

Question 1.

5/22 = 22.73% that I am a wolf
1/11 = 9.09% that I am a defender
1/22 =4.55% that I am a seer

Question 2.

Two independent events are:
Me being a defender and me owning a taco farm on the world map
Two dependant events are:
Aerso being a wolf and Aerso being a defender

Question 3.

Let S be the probability of being the seer.
Let L be the probability of being lynched.
P(S) = P(L) = 1/15
P(Not(S) and L) = P(Not(S)) * P(L)
P(Not(S) and L) = (1 - P(S)) * P(L)
P(Not(S) and L) = (1- 1/15) * (1/15)
P(Not(S) and L) = 0.9333 * 0.0667
P(Not(S) and L) = 6.23%

Some bad argumentation to get the ball rolling:
If there's a 5% chance that Aerso is the Mayor and not a werewolf and he is the Mayor, we ought to lynch him because there's a 95% chance he's the Mayor and a werewolf.

[Sorya]

SPOILER: Click to show

{Half Eaten Homework (I blame the swans)}
Question 1.
A werewolf game has 22 players including 5 wolves, 2 defenders and 1 seer. What is your probability of being assigned a wolf role? What is your probability of being assigned a defender role? What is your probability of being assigned the seer role?

The probability of being a wolf would be 5/22 while defender would be 1/11 and seer 1/22.



Question 2.
Give an example of two independent events, and two dependent events.


A dependent event would be being chosen as the mayor and later talking to Elite while dependent events would be lynching two wolves in a row.


Question 3.
There is a werewolf game with 15 players and 1 seer. In the first phase, one player will be lynched. We will assume that being the seer and being lynched are independent, and each player has an equal chance of being the seer or lynched. What is the probability that Aersoldorf is not the seer, but is lynched on the first day?

Aerso has a 1/15 chance of being the seer as well as a 1/15 chance in being lynched. Since he isn't the seer, his chances are 14/15 added to his being lynched 1/15 means that the odds are 14/225.

Discussion 1.
In the example, Aersoldorf had a 5% of being elected Mayor and not being a Wolf. If he is elected Mayor, should we trust him or lynch him? What about if he is not elected Mayor?

A 5% chance of being elected Mayor and not being a wolf leads me to believe there is a 95% chance that he is either elected and a wolf, not elected and a wolf, and not elected and not a wolf. If he is elected, I would most certainly not trust him but, if he went unelected, it is almost a coin flip. I would likely go towards trust since there are usually more players than there are wolves so the odds of him not being in a position of power and being a wolf are lower than those of him not being in a position of power and being one of the few wolves.

[Hahiha / USSR]

SPOILER: Click to show

Homework

Question 1.
A werewolf game has 22 players including 5 wolves, 2 defenders and 1 seer. What is your probability of being assigned a wolf role? What is your probability of being assigned a defender role? What is your probability of being assigned the seer role?

Let P(W) be the probability of a player being a wolf.
Let P(D) be the probability of a player being a defender.
Let P(S) be the probability of a player being the seer.

P(W)=5/22
P(D)=2/22=1/11
P(S)=1/22

Question 2.
Give an example of two independent events, and two dependent events.

Independent event A is the probability that I go to sleep before 22:00 (not very likely).
Independent event B is the probability that I binge-watch Star Trek (very likely).

Dependent event A is the probability that I'll drink orange juice when I wake up, it is dependent upon dependent event B.
Dependent event B is the probability that I'll drink apple juice when I wake up, it is dependent upon dependent event A.

Question 3.
There is a werewolf game with 15 players and 1 seer. In the first phase, one player will be lynched. We will assume that being the seer and being lynched are independent, and each player has an equal chance of being the seer or lynched. What is the probability that Aersoldorf is not the seer, but is lynched on the first day?

Let P(S) be the probability of Aerso being the seer.
Let P(L) be the probability of Aerso being lynched on Day 1 (assuming that there is no no-lynch option)

P(S)=1/15
P(L)=1/15

P(not(P(S))*P(L))=(1-(1/15))*(1/15)
=(14/15)*(1/15)
=(14*1)/(15^2)
=14/225=0.054901=5.49%

Discussion 1.
In the example, Aersoldorf had a 5% of being elected Mayor and not being a Wolf. If he is elected Mayor, should we trust him or lynch him? What about if he is not elected Mayor?
Let P(W) be the probability of Aerso being a wolf.
Let P(M) be the probability of Aerso being elected Mayor.

P(W)=3/14
P(M)=1/14

P(P(W)*P(M))=(3/14)*(1/14)
=0.01530612244=1.5%
P(NP(W)*P(M)=(11/14)*(1/14)
=0.05612244897=5.6%
P(P(W)*NP(M)=(3/14)*(13/14)
=0.19897959183=19.9%
P(NP(W)*(NP(M)=(11/14)*(13/14)
=0.72959183673=73.0%

I'm going to postpone by arguments for when I'm more awake.

Note: due to me being tired and misunderstanding the question I accidentally determined that sqrt(-1)=3/14. You're all welcome for this breakthrough in maths, with this the US will colonize Mars way before the Russians. If you ask kindly I may show you the infallible maths that determined this.

[Woonsocket]

Posting this so I can get notifications when someone posts here. I will do homework and post later.

[Flag Flayers]

SPOILER: Click to show

{Spoiler}
A werewolf game has 22 players including 5 wolves, 2 defenders and 1 seer. What is your probability of being assigned a wolf role? What is your probability of being assigned a defender role? What is your probability of being assigned the seer role?

Firstly I'll name the events:

'A' will be the probability to be a wolf
'B' will be the probability to be a defender; and
'C' will be the probability to ba a seer

Since the probability of an event occouring is the amount of ways or times an event can occour devided by the total number of possibilities this can be easily solved.

P(A)=5/22
P(A)=0.227
P(A)=22.7%

P(B)=2/22
P(B)=0.091
P(B)=9.1%

P(C)=1/22
P(C)=0.045
P(C)=4.5%

Give an example of two independent events, and two dependent events.

an example of two independent events would be two events where the result of one event doesn't affect the result of the other,
such as in this case:

I am drinking water and Tanzoria playing football.

Either if I am drinking or not drinking water, it will not affect the probability of Tanzoria playing football, vice versa. (Unless some crazy chaos theory, butterfly effect crap is going to happen ._.)

on the other hand, two dependent events are events where the result of one event has an effect on the probability of the other event,
such as in this case:

I haven't drank water all day and I am dehydrated.

If I haven't drank water all day, the probability of me being dehydrated is higher since being dehydrated means I lost a large amount of water. Vice versa, if I'm dehydrated it is more likely I haven't drank all day which is what cause the dehydration in the first place.

There is a werewolf game with 15 players and 1 seer. In the first phase, one player will be lynched. We will assume that being the seer and being lynched are independent, and each player has an equal chance of being the seer or lynched. What is the probability that Aersoldorf is not the seer, but is lynched on the first day?

Once again, first I'll label the different events:

'A' - Probability of being a seer
'B' - Probability of being lynched

I can then calculate the probability of both events:

P(A)=1/15
P(A)=0.066...
P(A)=6.67%

P(B)=1/15
P(B)=0.066...
P(B)=6.67%

Since we are asking what is the probability of Aersoldorf NOT being the seer we need to calculate it first

P(Not A)=1-P(B)
P(Not A)=14/15
P(Not A)=0.933...
P(Not A)=93.33%

To combine two independent probabilities all we need to do is multiply the two probabilities together

P(B And Not(A))=1/15*14/15
P(B And Not(A))=14/225
P(B And Not(A))=0.062...
P(B And Not(A))=6.22%

Therefore, the probability of Aersoldorf not being the seer and being lynched on the first day is 6.22%

{Spoiler}

Discussion:
In the example, Aersoldorf had a 5% of being elected Mayor and not being a Wolf. If he is elected Mayor, should we trust him or lynch him? What about if he is not elected Mayor?

If Aersoldorf had a 5% probability of being elected Mayor and not being a Wolf it doesn't give us much to go on,
it only means that he has a 95% chance of:

Being a wolf and not being mayor
Being a wolf and being a mayor
Not being a wolf and not being a mayor

From the example I know that Aersoldorf has a probability of 3/14 to be a wolf and 1/14 to be mayor.
To assume if he's trustworthy we can compare his probability of being elected mayor and not being a wolf, to being elected as mayor and being a wolf.

If the probability of Aersoldorf being a wolf is 'A' and the probability he is elected mayor 'M', the probability of both events happening is 3/14*1/14, which is 0.0153, or 1.53%

The probability of Aersoldorf being elected mayor and being a wolf is a lot lower then the probability of him being elected and not being a wolf, which I'm assuming it means that by being elected as mayor, he has a lower probability to be a wolf, I'm not sure what that says on the (in)dependency of these events but according to the raw data, I'd trust him more.

Now if he wasn't elected mayor he would either:

Be a wolf and not be elected mayor
Not be a wolf and not be elected mayor

Doing all the calculation to compare the 2 probabilities, Aersoldorf has a probability of 19.9% to be a wolf while not elected as mayor, and has a probability of 72.95% of not being both a wolf and a mayor.
With a 20% probability of being a wolf, I'd be a lot less willing to trust him, even though he has a higher probability of being innocent, only because the probability of him being a wolf is too high for me to consider 'safe'.

These are at least my thoughts on that

Also I don't know how to make spoilers that look nice, or even make them correctly :l
So if anyone's feeling up to teach me how, you are welcome to do that.

[Alkasia]

SPOILER: Click to show

1.

wolf role- 5/22 = 22.7%

defender role- 2/22 = 1/11 = 9.1%
seer role- 1/22 = 4.5%



2.

Independent

I am the seer.

I am a Senator in XKI.



Dependent

The seer has discovered my role.

The seer will out me publicly.



3.

Let S be the probability of Aerso being the seer
Let L be the probability of Aerso being lynched on the first day


P(Not (S) and L)= (1- P(S))*(P(L))

P(S) = P(L)= 1/15

P(Not (S) and L)= (1-(1/15))*(1/15) = 14/225 = 6.22%

Discussion 1
Well let's see here:
Aerso is elected Mayor and is not a wolf- 5.61%
Aerso is elected Mayor and is a wolf- P(M and A)= (3/14)*(1/14)= 1.53%
Aerso is not elected Mayor and is a wolf- P(Not (M) and A)= (1-(1/14))*(3/14)= 19.9%
Aerso is not elected Mayor and is not a wolf- P(Not M and A)= (1-(1/14))*(1-(3/14))= 73%

As a general rule, nobody should be trusted starting out, but if we're deciding on Aerso and he's elected Mayor, 1.53% chance of being a wolf definitely does not warrant a lynching, especially when not being Mayor and being a wolf provides around a 20% chance of being a wolf. If he is not elected Mayor, that leaves him in the same boat as almost everyone, so lynching him is as good a bet as any, and is definitely less trustworthy. It's really chance at that point.

[Palman]

SPOILER: Click to show

Question 1.
A werewolf game has 22 players including 5 wolves, 2 defenders and 1 seer.
What is your probability of being assigned a wolf role?
What is your probability of being assigned a defender role?
What is your probability of being assigned the seer role?

5/22 chance of being a wolf -> 22.73
2/22 chance of being a defender -> 9.09%
1/22 chance of being the seer -> 4.55%


Question 2.
Give an example of two independent events, and two dependent events.

Independent: Hakke is lynched first and Ater is running the game.
Dependent: Hakek is lynched first and Louistan is lynched first. (whether or not Hakke is lynched impacts the odds that Louis is lynched)


Question 3.
There is a werewolf game with 15 players and 1 seer. In the first phase, one player will be lynched. We will assume that being the seer and being lynched are independent, and each player has an equal chance of being the seer or lynched. What is the probability that Aersoldorf is not the seer, but is lynched on the first day?

1/15 chance that Aerso is the seer, thus 14/15 chance that he is not the seer.
1/15 chance that he is lynched.
So, the chances that he is lynched (1/15) times the odds he is not the seer (14/15) gives us 14/225 or 6.22%.

Discussion 1.
In the example, Aersoldorf had a 5% of being elected Mayor and not being a Wolf. If he is elected Mayor, should we trust him or lynch him? What about if he is not elected Mayor?

So, I think everyone's making the basis of the discussion question more difficult than it is. We're told that being the Mayor and being a wolf are independent. So when we're asked if we should trust him, whether or not he's the mayor is immaterial. Regardless of his mayorship, there is a 3/14 -> 21.43% chance he's a wolf.

We can actually calculate that from the 5-point-whatever percent, but I suspect Ater Nox doesn't want to go into the Conditional Probability equation yet.

There's another sticky factor, that I don't see anyone considering: What role are we in this case? If we're a wolf, the question is trivial; we already know if Aerso's a wolf and whether we can trust him. But say we're not a wolf. Then there are only 14 roles unaccounted for, and I'd argue that Aerso's odds are 3/13 to be a wolf. Odds would be 23.08%, not 21.43%. Small, but as the game goes on, that difference becomes more significant.

Updated: sorry thought it said 15 players not 14.

[Free Guns For All]

SPOILER: Click to show

Q1
P(w) = 5/22 (22.72%)
P(D) = 2/22 (9.09%)
P(S) = 1/22 (4.54%)

Q2
Independent: (1) Ater Nox is a wolf and (2) The game's host is Boltor
Dependent: (1) Ater Nox is a wolf and (2) Ater Nox is voted the alpha wolf (a role I made up for the purposes of this class)

Q3
P(L) and P(S) = 1/15 (6.6%)
P(Not L) and P(Not S) = 14/15 (93.3%)
P(L and Not(S))
P(L and Not(S)) = P(L) * P(Not(S))
P(L and Not(S)) = P(L) * (1 - P(S))
P(L and Not(S)) = 1/15 * (1 - 1/15)
P(L and Not(S)) = 0.06 * (1 - 0.06)
P(L and Not(S)) = 0.062 (6.2%)

D
Regardless of whether he's elected mayor, it is impossible to trust him completely until there is concrete evidence for or against him. Just because it's only a 5% chance of our mayor not being a wolf doesn't make it impossible, just a lot less likely. Plus, as Palman brought up, if you are a wolf, you would already know whether to trust him or not.