Task 1. On a plate, there are various candies of three types: 2 lollipops, 3 chocolate candies, and 5 jelly candies. Svetlana sequentially ate all of them, choosing each subsequent candy at random. Find the probability that the first and last candies eaten were of the same type.
Answer: $14 / 45$.

Solution. Two candies of the same type can be either lollipops, chocolates, or jellies. We will calculate the probabilities of each of these events and add them up.

Arrange the candies in the order of their consumption. The probability that the first candy is a lollipop is $2 / 10$. The probability that the last candy is a lollipop is the same as the probability that the lollipop is in any other position. Therefore, this probability is $1 / 9$, since after choosing the first candy, only 9 candies remain, among which there is exactly one lollipop. Thus, the probability that the first and last candies are lollipops is $2 / 10 \cdot 1 / 9=2 / 90$.

Similarly, we find the probability that the first and last candies are chocolates, which is $3 / 10 \cdot 2 / 9=6 / 90$. The probability that the first and last candies are jellies is $5 / 10 \cdot 4 / 9=20 / 90$. Therefore, the answer to the problem is the number

$$
2 / 90+6 / 90+20 / 90=28 / 90=14 / 45 .
$$

Remark. The probability that the first and last candies are lollipops can also be calculated as follows.

There are $\frac{10!}{2!\cdot 3!\cdot 5!}$ ways to arrange our 10 candies in a row, and among them, there are $\frac{8}{3!\cdot 5!}$ ways to arrange them in a row so that the lollipops are at the beginning and the end. Then the probability that the first and last candies are lollipops is

$$
\frac{\frac{8}{3!\cdot 5!}}{\frac{10!}{2!\cdot 3!\cdot 5!}}=\frac{2}{90}=\frac{1}{45}
$$

Similarly, the probabilities for candies of other types can be calculated.

## Criteria

7 points. Any complete solution to the problem.

In a solution similar to the author's but containing arithmetic errors, the following criteria are summed:

2 points. The probability that the first and last candies are lollipops is correctly found.

2 points. The probability that the first and last candies are chocolates is correctly found.

2 points. The probability that the first and last candies are jellies is correctly found.

1 point. The correct answer is found.