## Task A-4.1.

Girls Maria and Magdalena are playing a chess match consisting of three games. The probabilities that Maria will win, lose, or draw in a single game are equal. The overall winner of the match is the girl who wins more games (out of three), and if they have an equal number of wins, the match ends in a tie.

What is the probability that Maria will be the overall winner of the match?
## First Solution.

The probability of each event (Magdalena's victory, Maria's victory, or a draw) is $\frac{1}{3}$.

The duel will end with Maria's victory in the following cases:

- Maria has won all three games;
- Maria has won exactly two games;
- Maria has won one game, and the other two have ended in a draw.

In the first case, the probability is $\frac{1}{3^{3}}=\frac{1}{27}$.

In the second case, we can choose the game in which Maria did not win in three ways, and then we have two choices for that game, either a draw or Magdalena's victory. Therefore, the probability in this case is $\frac{6}{27}$.

In the third case, we can choose the game in which Maria won in three ways. Therefore, the probability in this case is $\frac{3}{27}$.

Thus, the total probability of Maria's victory is $\frac{1}{27}+\frac{6}{27}+\frac{3}{27}=\frac{10}{27}$.