Conditional Probability

[shadow niche]

I need to find the most accurate probability P(R)

I am given events Y, G, B, O, and T which are independent but can also occur at the same time...

I have probabilities P(Y), P(G), P(B), P(O), P(T)
and probabilities P(R|Y), P(R|G), P(R|B), P(R|O), P(R|T)

how do I properly define P(R) if it is dependent on events Y, G, B, O, and T

I need to define a general case for P(R) given any number of the events could occur at the same time or not occur with the probabilities given...

[near pasture]

Conditional Probability (Please help lol)

Conditional Probability

[patent token]

@shadow niche have you started somewhere about the problem?

[shadow niche]

I have been researching Bayesian Networks because this scenario is just a basic example of all of the probabilities I’m trying to work with in a project I’m programming in Swift and I’m hoping I can learn how to properly implement it but I first need to understand the math going on and I just genuinely don’t know how to solve this or like how to even call the problem cause it doesn’t make sense to me to use total probability since the events can occur at the same time and so they span more than the sample space I would assume…

[patent token]

hmmm i see

alright, think of it this way;
R could only happen if any of the events that it depends upon happens, yes?

so u wud have to assume cases of the events happening
suppose in a single throw, only 'Y' occurred...Now, for 'R' to occur after 'Y' has occurred, is governed by P(R|Y)

now suppose in a single throw, 'Y', 'T' and 'O' occurred, what was the probability of them occuring together? Now, after they've occured together, what's the probability of 'R' occurring in 'atleast' one of these cases?

I must say that the number of cases you'll be getting would be pretty huge

try thinking of simpler cases first
suppose some event A occurs with a probability of 1/3
an event C dependent on A occurs with a probability of 1/4 after A has happened
notice the difference between the probability of even C happening
and the probability of C happening after A has happened

[shadow niche]

Yah I understand the basics of Bayes Theorem I’m just getting a little bit stuck on how to map this massive amount of cases to just get a general definition of P(R)

[elfin orbit]

Can you be more specific as to what Y G B O and T are ?

Two independent events cannot be mutually exclusive anyway

[shadow niche]

Well I just decided to make some assumptions to remove interdependency because I'm calculating probabilities in a trick-taking card game, Rook, so the probability i'm trying to start with is just the probability that a player has a given color based on the perspective of another player. I have all the functions set up to update these probabilities based on a simple hypergeometric distribution. The probability of a player having a color is interdependent on whether they have other colors and whether other players have the colors and not (because they are forced to follow suit, I believe this dependency is created). So I am just ignoring the interdependency on colors and I'm just calculating to retrieve the best likelihood of having the color I'm considering. I tried to iterate the interdependency to find convergence but I didn't really know what I was doing there and hence why I am just assuming the probability of other colors can be independent for the sake of this calculation.

So Y is the probability the player being considered is out of Yellow

G is the probability out of Green

B is the probability out of Black

O is the probability that 1 of the remaining two players is out of Red (there are four players in total, and I'm already accounting for a player in whose perspective is being used to calculate the probability then theres the player of interest so two players remaining)

Then T is the probability that both the remaining two players are out of Red

[shadow niche]

So What I'm trying to get in the end is R, the probability that the player is out of Red, and I have all the calculations for what I mentioned but I am not sure how to combine it or if there is an easier way to deal with interdependencies or just the best approach to keep track of these probabilities dynamically in my code (This is one of the simplest probabilities I have to calculate)

[elfin orbit]

I'm afraid this way of framing it won't cut it

I just read the rules of the game

You can't just ignore the other decks

Otherwise you probably will not derive meaningful results

I don't know exactly how you could approach this best, but my best guess would be to attempt to implement Monte Carlo simulations

And see what you can find

Perhaps you can find some literature about shuffling card game simulations

Obviously there are constraints, not all decks are possible

And not all positions can happen

[granite forum]

A good starting point would be just counting the cards in each suit and noting when someone is out, then the remaining cards of a given suit are in the hands of players who aren't out or in the undistributed cards.

from there you can build into more complicated understanding of the game and asking questions like given strategy S, what is the probability that a player has hand X given the choices they made in the past.

[shadow niche]

Yah that’s what I’ve been doing since yesterday I just started coding it to calculate the probabilities based on just knowing the remaining cards that are out and ignoring how the likelihood of a different player being out of the color influences the likelihood that they are I’ve just done if the other player is known to be out of the color how does that then change the probability otherwise it keeps a generally evenly distributed probability between each player having a card without more information which they im just feeding in extra information like that to try and improve it, this is the most accurate I know how to be with the knowledge I have of probability and statistical models/networks