# Simplify difference of two probabilities

Pr(Y_ij≥Y_ik+b_j−b_k ∀k)−Pr(Y_ij≥Y_ik ∀k) ?

I want to keep things as general as possible.

Y_ij is the outcome if entity i chooses action j, Y_ik is the outcome if entity i choose action k

b_j is the cost of doing j, b_k is the cost of choosing action k

These are discrete random variables. For illustrative purposes, can suppose k ranges from 1-10 i.e. 10 different actions to choose from.

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You should add more details. What are $Y_{ij}$ and $b_j$.? Are these discrete random variables? What is the range of $k$? You should also offer a good bounty for questions at this level, otherwise users may not have incentive to spend time on your question.

Thanks for this advice and I updated the question to add more detail. What is a fair bounty for this type of question? :)

Think about how much time one needs to spend to answer your question, and what a fair hourly rate for such individual would be. It will give you a pretty good idea. Keep in mind that people are busy and rarely decide to spend their time to answer somebody else's question unless there is a good incentive.