Random Variable - Birthday Problem

For question 1, you are almost there.

The argument goes that $3$ is not enough to have the probability that at least two of them were born on the same day of the week at least $\frac12$, but $4$ is, since

• $1-\frac77\times\frac67\times\frac57\qquad \approx 0.39 <\frac12 \text{ when } n=3$
• $1-\frac77\times\frac67\times\frac57\times\frac47 \approx 0.65 \ge\frac12 \text{ when } n=4$

Your $P_n = \frac{7-(n-1)}{7} P_{n-1}= \frac{8-n}{7} P_{n-1}=\frac{7!}{(7-n)!\,7^n}$ gives the probability that no two of them were born on the same day of the week, obviously starting with $P_0=P_1=1$, though the question asks for $1-P_n$