A recent survey of post-secondary education students in Canada revealed that $73$% know what type of job they want, when they graduate. You are to randomly pick $32$ post-secondary education students across the country, and ask each the following question: Have you selected a particular career path?
You have defined the random variable $X$ to represent the number, out of 32 post-secondary students chosen, who responded YES
If you continued to select additional students, what is the probability that the 50th student selected will be the 39th student to respond YES?
I am completely stuck on this question, and am not quite sure how to approach it. Any hints to help me on the path to solving this problem would be greatly appreciated.
We want $38$ yes's in the first $49$ surveys followed by a yes on the $50^{th}$
Note that this is a negative binomial with $n$ trials given $k$ successes where $n=50$ and $k=39$.
We have
$$\begin{align*} P(X=n) &={n-1 \choose k-1}p^k(1-p)^{n-k}\\\\ &={49 \choose 38}0.73^{39}\cdot0.27^{11}\\\\ &\approx0.07569 \end{align*}$$