Lesson 2

Let us show that when combining proportions, they can change counterintuitively. The effect described in the problem below is known as Simpson's paradox.

Problem. The faculty has two educational programs. This year, an equal number of girls and boys applied to the faculty, and each person applied to exactly one program. It turned out that in each program, the percentage of admitted girls was lower than the percentage of admitted boys. Show that it could happen that the overall percentage of admitted girls to the faculty was higher than the percentage of admitted boys.

To do this, find such positive integers \(n, w'_{1} \le w_{1}, m_{1}' \le m_{1}, w'_{2} \le w_{2}, m'_{2} \le m_{2}\) such that \[n=w_{1}+w_{2}=m_{1}+m_{2},\quad \frac{w_1'}{w_1}< \frac{m_1'}{m_1}, \quad \frac{w_2'}{w_2}< \frac{m_2'}{m_2}, \quad \frac{w_1'+w_2'}{w_1+w_2}> \frac{m_1'+m_2'}{m_1+m_2}.\] Here, \(n\) is the total number of girls who applied to the faculty, \(w_{1}\) is the number of girls who applied to the first program, and \(w_{1}'\) is the number of those admitted. The remaining parameters are defined similarly. Input the numbers \(n\), \(w_{1}'\), \(w_{1}\), \(m_{1}'\), \(m_{1}\), \(w_{2}'\), \(w_{2}\), \(m_{2}'\), \(m_{2}\) separated by spaces.