How many people must there be in a room before the probability that someone has the same birthday as you do is at least $1/2$? How many people must there be before the probability that at least two people have a birthday on July 4 is greater than $1/2$?

The probability of a person not having the same birthday as me is $(n-1)/n$. The probability of $k$ people not having the same birthday as me is that, squared. We apply the same approach as the text - we take the complementary event and solve it for $k$:

$$ 1 - \bigg(\frac{n-1}{k}\bigg)^k \ge \frac{1}{2} \\ \bigg(\frac{n-1}{k}\bigg)^k \le \frac{1}{2} \\ k\lg\bigg(\frac{n-1}{n}\bigg) \ge \lg\frac{1}{2} \\ k = \frac{\log(1/2)}{\log(364/365)} \approx 263 $$

As for the other question:

$$ \begin{align} \Pr\{\text{2 born on Jul 4}\} &= 1 - \Pr\{\text{1 born on Jul 4}\} - Pr\{\text{0 born on Jul4}\} \\ &= 1 - \frac{k}{n}\bigg(\frac{n-1}{n}\bigg)^{k-1} - \bigg(\frac{n-1}{n}\bigg)^k \\ &= 1 - \bigg(\frac{n-1}{n}\bigg)^{k-1}\bigg(\frac{n+k-1}{n}\bigg) \end{align} $$

Writing a Ruby programme to find the closest integer, we get 115.