# Forecasting Ball Colours: 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 II), grades 9–11. Answer and official solution.*

**Answer:** 1

**Solution.**

The next three one-minute trials are independent, and in each minute a blue ball comes out with probability $p = \frac{1}{3}$, regardless of the past six minutes. Vasya wrote “blue” in all three fields, so the number of his correct predictions is

$$X = X_1 + X_2 + X_3,$$

where $X_i$ is the indicator of the event “a blue ball came out in minute $i$”, that is, $\mathbb{P}(X_i = 1) = \frac{1}{3}$, $\mathbb{P}(X_i = 0) = \frac{2}{3}$.

By linearity of expectation,

$$\mathbb{E}X = \mathbb{E}X_1 + \mathbb{E}X_2 + \mathbb{E}X_3 = 3 \cdot \mathbb{P}(X_i = 1) = 3 \cdot \tfrac{1}{3} = 1.$$

So, on average, Vasya will guess $\boxed{1}$ of the 3 predictions correctly.