How to calculate the possibility of every scenario with respect to correlation? (Stochastic Optimization)

Question

Imagine having two variables with three possible outcomes each:

d_1 = [1, 2, 3] and probabilities p_1 = [0.5, 0.25, 0.25]
d_1 = [1, 2, 3] and probabilities p_2 = [0.5, 0.25, 0.25]

Now i can calculate the probability of every possible permutation like this:

scenario_1: d_1 = 1 and d_2 = 2 p(scenario_1) = 0.5 * 0.5 = 0.25

But if d_1 and d_2 are correlated the probability of 0.25 is not correct. Now my question: How do i correct this probability with respect to the correlation?

For example if corr[d_1, d_2] = 1

I feel like the probability of scenario_1 would have to be much higher.

Thanks!!

Answer 1

This depends on the kind of correlation definition you are using.

To compute the joint probability of two dependent variables A and B:

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

Provided the correlations are given, it should be possible to compute the conditional probabilities P(A given B) or P(B given A).