I have trouble getting a certain type of probability situation where the probability changes overtime. So I made up this problem, in a supermarket there are $150$ cartons of milk of which $50$ are spoiled. $150$ customers arrive one after the other and choose a carton of milk at random. Which place/customer in the queue has the most chance of getting an unspoiled carton of milk?
I hope this makes clear the situation. I thought maybe of using binomial distribution and expected value but how can I approach such a problem with probability change after every iteration? My question here ist not specifically about my mentioned problem but rather about a generalization of it. The first person has a probability of $\frac{100}{150}$ to get an unspoiled milk carton (U) and $\frac{50}{150}$ to get a spoiled one (S). But afterwards we get $\frac{99}{149}$ for U and $\frac{49}{149}$ for S, so how to get an expected value here? Is this conditional expectation?
We number the spoiled milk from $1$ to $50$ and the good milk from $51$ to $150$. Now let $\sigma(i)$ be the milk that the $i$th customer buys. Then $$ \Omega=S_{150} \Rightarrow |\Omega|=150!. $$ We define $$ A_i:=\text{,,customer } i \text{ chooses spoiled milk}"=\{\sigma\in\Omega\mid\sigma(i)\leq 50\}. $$ For fixed $1\leq k\leq 150$ holds $$ |\{\sigma\in\Omega\mid\sigma(i)=k\}|=149! $$ and thus $$ |A_i|=\sum_{k=1}^{50}|\{\sigma\in\Omega\mid\sigma(i)=k\}|=50\cdot 149! $$ The probability is therefore $$ P(A_i)=\frac{|A_i|}{|\Omega|}=\frac{50\cdot 149!}{150!}=\frac{50}{150}=\frac{1}{3}. $$ Therefore, everyone has the same chance.