What is the probability of getting a PhD position after getting some number of rejections?
If the chance of getting accepted at any given place is $p$, then the probability of getting accepted to at least one given $n$ applications is
$$ P = 1 – (1-p)^n$$
But receiving a rejection doesn’t just reduce the number of possible applications ($n$), it should also change our estimate of the probability of getting accepted to a single place $p$. Through the magic of maths and Bayes rule, the probability $p$ after $f$ failures is
$$p = \frac{\alpha_0}{\alpha_0 + \beta_0 + f}$$
Here $\alpha_0$ and $\beta_0$ are parameters which tell us what we think the probability $p$ should be before we’ve got any news, and how certain we should be about that value. More specifically $\alpha_0$ is a ‘pseudo count’ of how many (virtual) successes we’ve seen, and $\beta_0$ is how many (virtual) failures we’ve seen. If we want to be uncertain about the estimate we can make these small, so that new data will have a large effect. It seems pretty sensible for $\frac{\alpha_0}{\alpha_0 + \beta_0}$ to be equal to the our inital estimate of $p$. For example if I think $p\approx 10\%$, then I might set $\alpha_0=1$, $\beta_0=9$.
So the overall probability of being accepted to at least on position, given you applied for $n$ positions and have recieved $f$ rejections is
$$ P = 1 – \left(1-\frac{\alpha_0}{\alpha_0 + \beta_0 + f}\right)^{n-f}$$
If we apply for $n=6$ schools, and set $\alpha_0=1$, $\beta_0=9$, then the probability $P$ as function of $f$ looks like
Well that was more depressing than I initially thought. Here’s a very short colab notebook if you want to play with the numbers yourself.