Question

Одиннадцать детей встают в хоровод в случайном порядке. Среди них Антон и его сестра Маша.Какова вероятность того,что Антон и Маша окажутся рядом?

Answer 1

Пусть Антон будет первым встал в хоровод. Тогда рядом с Антоном есть два места, среди этих двух мест может оказаться Маша. То есть, Маша должна занять 2 из 10 оставшихся позиций рядом с Сережей. 

Всего благоприятных событий 2. А всего все возможных событий: 10

Вероятность того, что Маша окажется одной из десяти на одном из двух мест: $P= \dfrac{2}{10}=0.2$

Answer 2

Probability of Anton and Masha being together in the circle

To calculate the probability of Anton and Masha being together in the circle, we need to consider the total number of possible arrangements and the number of arrangements where Anton and Masha are next to each other.

Let's assume there are 11 children in total, including Anton and Masha. Since the children are standing in a circle, we can fix one of them (let's say Anton) and arrange the remaining 10 children in a line. This can be done in 10! (10 factorial) ways.

Now, we need to consider the number of arrangements where Anton and Masha are next to each other. Since they need to be together, we can treat them as a single entity. So, we have 10 entities to arrange, which include the combined entity of Anton and Masha. This can be done in (10-1)! ways.

Therefore, the probability of Anton and Masha being together in the circle is given by:

Probability = (Number of arrangements where Anton and Masha are together) / (Total number of possible arrangements)

Probability = [(10-1)!] / [10!]

Simplifying the expression:

Probability = 1/10

So, the probability of Anton and Masha being together in the circle is 1/10.

Please note that the above calculation assumes that all arrangements are equally likely and that there are no other constraints or preferences mentioned in the question.