4.5.2 problem 71

Let $I_j$ be the indicator random variable for pair $j$ having the aforementioned property. $P(I_j=1)=\frac{1}{365^2}$, under the assumption that the probability of being born on a particular day is $\frac{1}{365}$. Note that since we don’t know anything about the age of the kids, we are assuming their mothers are also equally likely to be born on any of the $365$ days.

Then, the expected number of pairs with the aforementioned property is $\text{E}\left(X\right)=\left(\frac{90}{2}\right)\frac{1}{365^2}\approx 0.03$.

Let $Z\sim \text{Poiss}(0.03)$ model the distribution of pairs with the desired property. Then, probability that there is at least one such pair is $1-P(Z=0)=1-e^{-0.03}\approx 1-(1-0.03)=0.03=\frac{3}{100}$.