Bank "Imperial" charges a fee for withdrawing money from an account, which consists of two parts: a fixed payment for processing the transaction and an additional payment proportional to the amount withdrawn. For example, when withdrawing 5000 rubles, the depositor pays 110 rubles, and when withdrawing 11000 rubles, the depositor pays 230 rubles. How much will the depositor pay in fees if they want to withdraw 8000 rubles?
8000 is the arithmetic mean of the numbers 5000 and 11000. Both parts of the commission are also "averaged" when calculating the arithmetic mean. Therefore, the required commission is $1 / 2(110+230)=170$.

## Answer

170 rubles.

In a school football tournament, 8 teams of equal skill participate. Each game ends with one team winning. A randomly drawn number determines the team's position in the tournament table:

![](https://cdn.mathpix.com/cropped/2024_05_06_53798a1e86d520071242g-15.jpg?height=783&width=1384&top_left_y=217&top_left_x=337)

What is the probability that teams $A$ and $B$:

a) will meet in the semifinal;

b) will meet in the final.

## Solution

a) For teams to meet in the semifinal, they must be placed in different but converging subgroups (event $X$).

Team $A$ can be placed in any subgroup. For team $B$ to meet team $A$ in the semifinal, it must be placed in an adjacent subgroup.

The probability of this is $2 / 7$.

Then both $A$ and $B$ must win their matches (event $Y$). Since the teams are equally skilled, the probability of winning in a subgroup is $1 / 2$, and the wins of teams $A$ and $B$ are independent events, so $P(Y)=1 / 2 \cdot 1 / 2=1 / 4$.

Since events $X$ and $Y$ are independent, the required probability is $P(X) P(Y)=2 / 7 \cdot 1 / 4=1 / 14$.

b) For teams to meet in the final, team $B$ must be placed in the half of the table where team $A$ is not. Accordingly, $P(X)=4 / 7$.

In addition, each team must win two matches. $P(Y)=(1 / 2)^{4}=1 / 16$.

The required probability is $P(X) P(Y)=4 / 7 \cdot 1 / 16=1 / 28$.

## Answer

a) $1 / 14$;
b) $1 / 28$.