# What is the probability of majority agreement?

Nine people are rating an object on a scale from 1 to 6.

What is the probability that at least five out of nine people (i.e. a majority) will agree (i.e. provide the same rating, whether it is 1, 2, 3, 4, 5, or 6)?

EDIT: I am interested in pure chance agreement.

EDIT2: $$\sum_{r}^n \frac{(k-1)^{n-r}}{k^n}\binom nrk$$where n is the number of raters, r is the minimum frequency of agreement, and k is the number of categories.

Is this correct? Is this formula named?

• What have you tried? Sep 1, 2019 at 10:25
• Inclusion-exclusion might be useful here. Sep 1, 2019 at 11:43
• Are ratings selected uniformly at random (from $1,2,3,4,5,6$) by each rater? This seems unrealistic. If the ratings are not selected uniformly at random, then you would need more information about how ratings are assigned. Sep 1, 2019 at 13:08
• @paw88789 Unrealistic is correct. I want to compare the observed agreement I calculated against the agreement expected by chance. Sep 1, 2019 at 18:27
• @ShubhamJohri My answer is the following: (1/6)^5*(5/6)^4*126*6 + (1/6)^6*(5/6)^3*84*6 + (1/6)^7*(5/6)^2*36*6 + (1/6)^8*(1/6)*9*6 + (1/6)^9*6 = 0.05367893613 If this is correct, I am interested in a general formula. Sep 3, 2019 at 10:35