# 9. Solution.

1st method. An elementary outcome in the random experiment is a triplet of positions where the children in green caps stand. Consider the event $A$ "all three green caps are together." This event is favorable in 9 elementary outcomes. The event $B$ "two green caps are together, and the third is separate" is favorable in 45 elementary outcomes (9 ways to choose two adjacent places, and the third place must be one of the 5 places not adjacent to the already chosen ones). The total number of ways to choose a triplet is $C_{9}^{3}=84$. Then the probability of the event "some two green caps are together" is $\mathrm{P}(A)+\mathrm{P}(B)=\frac{54}{84}=\frac{9}{14}$, and the desired probability is $1-\mathrm{P}(A)-\mathrm{P}(B)=\frac{5}{14}$.

2nd method. Number the children in green caps and form pairs "green-red" such that a red cap is placed to the right of each green cap. To the right of the first green cap, any of the six red caps can be placed, to the right of the second, one of the five remaining, and to the right of the third, one of the four remaining red caps. This results in three pairs "green-red" and three separate red caps. There are exactly $6 \cdot 5 \cdot 4=120$ such groupings. Arrange these pairs and separate caps (a total of six elements) in a circle. This can be done in 5! ways, and no two green caps will be adjacent, as a red cap is located to the right of each green cap.

The total number of ways to arrange 9 children in a circle is 8!.

Thus, the desired probability is $\frac{120 \cdot 5!}{8!}=\frac{120}{8 \cdot 7 \cdot 6}=\frac{5}{14}$.

3rd method is similar to the 2nd method from the first variant. Other methods of solution are possible.
Answer: $\frac{5}{14}$.

| Evaluation Criterion | Score |
| :--- | :---: |
| Solution is complete and correct | 3 |
| The chosen method is clear from the solution, but the probability of the event "not all three together" is correctly found | 2 |
| The chosen method is clear from the solution, but only the probability of the event "some two not together" is correctly found | 1 |
| Solution is missing or incorrect | 0 |