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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?

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    $\begingroup$ What have you tried? $\endgroup$ Sep 1, 2019 at 10:25
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    $\begingroup$ Inclusion-exclusion might be useful here. $\endgroup$ Sep 1, 2019 at 11:43
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    $\begingroup$ 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. $\endgroup$
    – paw88789
    Sep 1, 2019 at 13:08
  • $\begingroup$ @paw88789 Unrealistic is correct. I want to compare the observed agreement I calculated against the agreement expected by chance. $\endgroup$
    – rabouillet
    Sep 1, 2019 at 18:27
  • $\begingroup$ @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. $\endgroup$
    – rabouillet
    Sep 3, 2019 at 10:35

1 Answer 1

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Hint-you need to select ,5 people out of 9, this 5 person can select any number out of 6 ,or you may think that no one get majority,i am not writing ,whole answer .

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