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The expected number of days of the year that are birthdays of exactly $3$people;

Define indicator random variables Ij that marks if that day is the birthday of exactly three people or not. Observe that

$P(I_j=1)=\left(\begin{array}{c}100\\ 3\end{array}\right){\left(\frac{1}{365}\right)}^3{\left(\frac{364}{365}\right)}^{97}$

since we have to choose a group of $3$people out of $100$them. The number of days in the year that satisfy this condition is $N=\underset{j=1}{\overset{365}{∑}}{I}_j$. Hence, the expected value is

$E\left(N\right)=\underset{j}{∑}E\left(I_j\right)=365â‹\dots \left(\begin{array}{c}100\\ 3\end{array}\right){\left(\frac{1}{365}\right)}^3{\left(\frac{364}{365}\right)}^{97}$

The expected number of days of the year that satisfy the condition is

$E\left(N\right)=\underset{j}{∑}E\left(I_j\right)=365â‹\dots \left(\begin{array}{c}100\\ 3\end{array}\right){\left(\frac{1}{365}\right)}^3{\left(\frac{364}{365}\right)}^{97}$

The expected number of distinct birthdays.

Define indicator random variables Ij that marks if there exists a person that has a birthday on that day or not. We have that

$P(I_j=1)=1−{\left(\frac{364}{365}\right)}^{100}$

The number of days in the year that fulfill this condition is $N=\underset{j=1}{\overset{365}{∑}}{I}_j$

Hence, the expected value of a distinct birthday is

$E\left(N\right)=\underset{j}{∑}E\left(I_j\right)=365â‹\dots (1−{\left(\frac{364}{365}\right)}^{100})$

The expected number of distinct birthdays that satisfy the condition is

$E\left(N\right)=\underset{j}{∑}E\left(I_j\right)=365â‹\dots (1−{\left(\frac{364}{365}\right)}^{100})$