# Forecasting Task Types

*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. The official answer and solution are in a separate file.*

At the qualifying stage of an online olympiad, each participant is given three problems, each of which is either in mathematics or in artificial intelligence. Suppose that in the new season the problem type at position $i$ coincides with last year's with probability $s_i$ and switches to the opposite with probability $1 - s_i$. The choice is made independently for different positions. It is known that $s_1 = 2/3, s_2 = 1/3, s_3 = 3/4$. Petya has analysed last year's problem sets and is sure that, even before the round starts, he can predict the problem types at all three positions at once: either none of them will change and they will be the same as last year, or all of them will switch to the opposite (that is, artificial intelligence will come up instead of mathematics, and vice versa).

Petya wants the average number of correctly guessed positions (the mathematical expectation) to be as large as possible. Which prediction is optimal in this case?

1. The problem types will not change
2. The problem types will switch to the opposite

Find the average number of positions (the mathematical expectation) guessed correctly by Petya if he follows the optimal strategy.

**Scoring criterion:** 6 points for each correct answer. Total — 12 points

**Maximum score for the task — 12**