Probability- Birthday problem

Question

The traditional way of finding the answer is to find the complement and subtract from one. I tried an alternative solution. I come up with a formula.

for $k$ people,
$P(k)=P(k-1)+{(k-1)\over365}\times{364\over365}\times{363\over365}\times...\times{{{365-(k-1)}}\over365}$
$P(k)=$Probabiltiy of match in $k$ people.

For example,
$P(2)={1\over365}$
$P(3)=P(2)+{2\over365}\times {364\over365}$
$P(4)=P(3)+{3\over365}\times {364\over365}\times {363\over365}$
and so on.

But, I cannot interpret the formula. I am studying probability for preparing to join my master's in data science. I am not an expert in this topic. I found this formula, somebody might have found it already. I want you to confirm that the formula is right. If it is right, what could be inferred from it.

Answer 1

Your formula is correct, and says that the probability of a duplicate birthday in the first $k$ people is the probability of a duplicate birthday in the first $k-1$ people plus the product of the probability of no duplicate birthday in the first $k-1$ people and the probability of the $k$th person duplicating one of the first $k-1$ people's birthday.

In effect you are saying $$P(k)=P(k-1) +\frac{k-1}{365}(1-P(k-1))$$ starting with $P(1)=0$.

This is not much change from what you call the traditional approach. If $Q(k)$ is the probability of no duplicate birthday in the first $k$ people then $Q(k)=1-P(k)$ and that equation is equivalent to $1-Q(k)=1-Q(k-1) +\frac{k-1}{365}Q(k-1)$ , i.e. $$Q(k)=\frac{365-k+1}{365}Q(k-1)$$ starting with $Q(1)=1$. This is what you call the traditional approach, and your reformulation is essentially similar. You can use whichever you prefer, but you should not expect to infer different things, except perhaps that with your reformulation it is even more obvious that $P(k)$ is an increasing function of $k$.