In a quiz contest,probabilities of teacher,boy students and girl students answering the question correctly are $\alpha,\beta$ and $\gamma$ respectively.The probability of teacher and student agreeing to the same answer is $\dfrac12$.Find the ratio of the number of boy students to the girl students.
There arise two cases when they agree-
(1) The teacher and the student are correct.
(2) The teacher and the student are wrong.
Case (1) has two subcases-
(i) The teacher and the male student are correct.
(ii) The teacher and the female student are correct.
Case (2) has two subcases-
(i) The teacher and the male student are wrong.
(ii) The teacher and the female student are wrong.
From this we can get,
$\alpha \beta+(1-\alpha)(1-\beta)+\alpha \gamma+(1-\alpha)(1-\gamma)=\dfrac12$
Now, I consider that there are total $n$ students and I consider there are $k$ male students. I need to find out the ratio $k:n-k$. But I cannot proceed further. Please help.