I have the formula for the probability of chance agreement where:

k people are giving ratings

n is the number of possible ratings to choose from

This is the probability for chance agreement where the ratings are allowed to differ by 2 (eg. 1 and 3 would be classed as in agreement).

I is an indicator function,

Can anyone help me describe how this probability occurs. I was hoping for an explanation along the lines of, "No. of exact agreements + No. of agreements where ratings differ by 1 ..." etc.

I understand the way that the probability for chance agreement in the case where the ratings can differ by one works:

The final term in the numerator is then the number of exact matches.

The (n-1) gives the number of adjacent pairs that would be classed as in agreement.

The summation can then be written as, which is the matches differing by one minus the exact matches as they have already been counted.

I just can't work out the similar ideas for the probability where ratings can differ by 2!