This is Blitzstein's Introduction to Probability (2019 2 ed) Ch 3, Exercise 8, p 141.
There are 100 prizes, with one worth $1, one worth $2, . . . , and one worth $100. There are 100 boxes, each of which contains one of the prizes. You get 5 prizes by picking random boxes one at a time, without replacement. Find the PMF of how much your most valuable prize is worth (as a simple expression in terms of binomial coefficients).
From my point of view the probability is equal to:
$P(X=k)$=$\binom{5}{1}$/$\binom{100}{5}$ for $k=1,2,…100$.
Because there are in total $C(100,5)$ ways of choosing the box and $C(5,1)$ of choosing the most valuable prize. What's wrong with my reasoning?
Best Answer
You are searching for probability that $k$ is the most valuable prize you chose. That is probability that all prizes you chose are less valuable than $k+1$ and that you chose $k$, which means that not all prizes are less than $k$.
Denote by $Y_i$ event that all 5 prizes you chose are less than $i$.
Then: $$P(Y_i)=\frac{i \choose 5}{100 \choose 5}$$ and: $$P(X=k)=P(Y_{k+1}\cap \neg Y_k)=P(Y_{k+1})-P(Y_{k+1} \cap Y_k)=P(Y_{k+1})-P(Y_k)=\frac{{{k}\choose{5}}-{{k-1}\choose{5}}}{{100}\choose{5}}$$