The probability of twins being born in Shvambria is $p$, and triplets are not born in Shvambria.

a) Estimate the probability that a Shvambrian met on the street is one of a pair of twins?

b) In a certain Shvambrian family, there are three children. What is the probability that there is a pair of twins among them?

c) In Shvambrian schools, twins are always enrolled in the same class. There are a total of $N$ students in Shvambria.

What is the expected number of pairs of twins among them?
a) Let $M$ pairs of twins and $N$ "singletons" walk the streets of the Shvambra cities and villages. Then $\frac{M}{M+N} \approx p$, from which $\frac{M}{N} \approx \frac{p}{1-p}$. If we meet a random Shvambra resident, the probability that they are one of the twins is

$$
\frac{2 M}{2 M+N}=\frac{2 \frac{M}{N}}{2 \frac{M}{N}+1} \approx \frac{2 \frac{p}{1-p}}{2 \frac{p}{1-p}+1}=\frac{2 p}{p+1}
$$

b) The desired probability is conditional. Let's draw a graph (red edges indicate the birth of twins, blue edges indicate singletons). The graph can be extended indefinitely downward, but we are only interested in the fragment that accounts for all possible ways of having three children (see figure). Next to each edge, we will write the probability. We will represent the event $A=\{$ Three children $\}$ with a curved area. This event includes three vertices, i.e., three elementary events with a total probability of $2 p q + q^3$.

![](https://cdn.mathpix.com/cropped/2024_05_08_293b4772a418c01a5915g-23.jpg?height=472&width=686&top_left_y=953&top_left_x=686)

Of these, the event $\mathrm{B}=\{$ There are twins $\}$ is favorable to two elementary events (chains containing red edges) with a total probability of $2 p q$. Therefore, the conditional probability

$\mathrm{P}(B \mid A)=\frac{\mathrm{P}(A \cap B)}{\mathrm{P}(A)}=\frac{2 p q}{2 p q + q^3}=\frac{2 p}{2 p + q^2}$

c) As we know, the probability that a randomly selected first-grader is one of a pair of twins is $2 p / (p+1)$. Let's introduce a random variable $I_{k}$ - the indicator of a twin: $I_{k}=1$ if the $k$-th first-grader is one of a pair of twins and $I_{k}=0$ if the $k$-th first-grader is a singleton. $\mathrm{E} I_{k}=2 p / (p+1)$.

The total number $S$ of pairs of first-graders who are twins is $1 / 2 (I_{1} + I_{2} + \ldots + I_{N})$.

Therefore, $\mathrm{E} S = \frac{1}{2} (\mathrm{E} I_{1} + \mathrm{E} I_{2} + \ldots + \mathrm{E} I_{N}) = \frac{1}{2} \cdot N \cdot \frac{2 p}{p+1} = \frac{N p}{p+1}$.

## Answer

a) $\frac{2 p}{p+1} ;$ b) $\frac{2 p}{2 p + q^2} ;$ c) $\frac{N p}{p+1}$.