1. To qualify for the competition, wrestler Vladimir had to conduct three bouts and win at least two in a row. His opponents were Andrey (A) and Boris (B). Vladimir could choose the sequence of matches: ABA or BAB. The probability of Vladimir losing a single bout to Boris is 0.3, and to Andrey is 0.4; the probabilities are constant. Which sequence gives a higher probability of qualifying for the competition, and what is this probability?
Answer. ABA; $p=0.588$.

Solution. Let Vladimir meet a weaker opponent twice, that is, the scheme is

![](https://cdn.mathpix.com/cropped/2024_05_06_88c5abeee9973c9a71c8g-3.jpg?height=54&width=1630&top_left_y=1332&top_left_x=293)
Vladimir chooses the ABA scheme. Then $p_{2}=0.6 \cdot 0.7 \cdot 0.4 + 0.6 \cdot 0.7 \cdot 0.6 + 0.4 \cdot 0.6 \cdot 0.7 = 0.588$.

Comment. Correct solution - 20 points. One arithmetic error in a correct solution - 18 points. Computational errors in a correct scheme - 15 points. It was not taken into account that wins must be in a row - 10 points. Some progress in the solution - 5 points. The solution is started, but the progress is insignificant - 1 point. The solution is incorrect or missing - 0 points. Only the correct answer without a solution - 0 points.