Thank you for this, it was a very interesting problem.
I might be a little confused by the indices n and t. If you're defining n as the total thieves and t as the other thieves being targeted, will the term (n-2 choose t-1) in the Tj equation resolve to 1 every time? For 10 thieves, will it always be 8 choose 8?
Tj is slightly nuanced random variable. If we assume we are thief (i) and we target thief (j), then Tj supposed to represent the random variable of the possible number of “other” thieves that target thief j. For example if one thief targets j (other than i) then Tj=1. If three thieves target j (other than i), Tj=3. With that in mind, lower case t is just an index variable over all possible number of other thieves that can target j so the variable spans from 0 to n-2. n is a constant and is always the total number of thieves in the game.
Thank you for this, it was a very interesting problem.
I might be a little confused by the indices n and t. If you're defining n as the total thieves and t as the other thieves being targeted, will the term (n-2 choose t-1) in the Tj equation resolve to 1 every time? For 10 thieves, will it always be 8 choose 8?
Tj is slightly nuanced random variable. If we assume we are thief (i) and we target thief (j), then Tj supposed to represent the random variable of the possible number of “other” thieves that target thief j. For example if one thief targets j (other than i) then Tj=1. If three thieves target j (other than i), Tj=3. With that in mind, lower case t is just an index variable over all possible number of other thieves that can target j so the variable spans from 0 to n-2. n is a constant and is always the total number of thieves in the game.