# Forecasting Ball Colours

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

An artificial intelligence club is holding a forecasting contest. There is a machine: every minute it produces a ball — either red with probability $\frac{2}{3}$ or blue with probability $\frac{1}{3}$. This happens independently: the colour of one ball does not affect the colour of the others. Before the contest began, the participants were told in what order the balls had come out over the last six minutes:

$$\text{blue, red, blue, blue, red, blue.}$$

According to the rules, three fields — the predictions for the next three minutes — must be filled in in advance and the sheet handed in. Vasya noticed that blue had occurred more often and wrote “blue” in all 3 fields. Find the mathematical expectation of the number of Vasya's correct predictions over the next three minutes, that is, how many of these three predictions will turn out to be correct on average?

**Scoring criterion:** exact match of the answer — 12 points

**Maximum score for the task — 12**