27.2. (England, 73). A teacher and students of a certain class are asked questions. The probability that the teacher's answer will be correct is $\alpha$, and the probability of a correct answer from a student is $\beta$ or $\gamma$ depending on whether a boy or a girl is answering, respectively. The probability that the answer of a randomly selected student will coincide with the teacher's answer is $1 / 2$. Find the ratio of the number of boys to the number of girls in the class.
27.2. Let $\delta$ denote the probability that a randomly chosen student will give the correct answer. The probability of the answer of a randomly chosen student coinciding with the teacher's answer is the sum of the probability $\alpha \delta$ of both giving the correct answer and the probability $(1-\alpha)(1-\delta)$ of both giving the wrong answer. Therefore, the condition of the problem can be written as

$$
\alpha \delta+(1-\alpha)(1-\delta)=1 / 2, \quad \text { or } \quad(\alpha-1 / 2)(\delta-1 / 2)=0
$$

If $\alpha=1 / 2$, then this condition is satisfied and the ratio of the number of boys to the number of girls in the class can be any. Suppose now that $\alpha \neq 1 / 2$. Then $\delta=1 / 2$. Let $x$ and $y$ denote the number of boys and girls in the class, respectively. The probability $\delta$ of a correct answer from a randomly chosen student in the class is the sum of the probability $(x /(x+y)) \beta$ that a boy will be chosen and he will give the correct answer, and the probability $(y /(x+y)) \gamma$ that a girl will be chosen and she will give the correct answer. Thus, the condition of the problem has the form

$$
\frac{1}{2}=\frac{x}{x+y} \beta+\frac{y}{x+y} \gamma, \text { or } \quad\left(\beta-\frac{1}{2}\right) x=\left(\frac{1}{2}-\gamma\right) y
$$

Therefore, if $\beta=\gamma=1 / 2$, then the ratio of the number of boys to the number of girls in the class can be any; if $\beta=1 / 2$, but $\gamma \neq 1 / 2$, then the class consists only of boys; if $\beta \neq 1 / 2$, then the ratio of the number of boys to the number of girls in the class is ( $1-2$ (of course, provided that the specified fraction is non-negative).