# Forecasting Task Types: Solution

*English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email sota.ai.community@gmail.com.*

*Task 2 of the school stage of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence (region group I), grades 9–11. Answer and official solution.*

**Answer:** The problem types will not change; 7/4

**Solution.**

Denote by $X_i$ the indicator of a correct guess at position $i$ (equal to 1 if Petya guessed correctly, and 0 otherwise). Then the average number of correct guesses is

$$E[X_1 + X_2 + X_3] = EX_1 + EX_2 + EX_3.$$

Two strategies:

- “As last year” (we predict that the types will not change): $EX_i = s_i$. Then $E = s_1 + s_2 + s_3 = \frac{2}{3} + \frac{1}{3} + \frac{3}{4} = \frac{7}{4}$.
- “The opposite” (we predict that all the types will switch): $EX_i = 1 - s_i$. Then $E = (1 - s_1) + (1 - s_2) + (1 - s_3) = \frac{1}{3} + \frac{2}{3} + \frac{1}{4} = \frac{5}{4}$.

Since $\frac{7}{4} > \frac{5}{4}$, it is optimal to predict “The problem types will not change”, and the maximum expectation equals $\boxed{\frac{7}{4}}$.