Problem: I played lottery and I bought $5000$ numbers of $100.000$. In the raffle, there are $2000$ prizes (or $2000$ tests), in that $2000$ prizes, $5$ are high prizes.
Question: What's my probability of get one high prize?
I used binomial with $5$ tests and $1$ success with probability of success $5%$. The problem of this approach is when you get a prize, the probability of success changes $(5000/99999)$
How to solve this? Am I doing anything wrong?
On
Using the binomial distribution for this problem is a very good approximation since the success probability is nearly the same for each of the 5 trials: $$p(n=1|N=5,p=0.05) \approx 0.203627$$ If you want to work this out exactly, add the probabilities of the mutually exclusive situations in which you win exactly once, namely, you win only the first draw $(p_1)$, or only the second draw $(p_2)$, or the third... $$p_1=\left({5000\over100000}\right)\left(1-{4999\over99999}\right)\left(1-{4999\over99998}\right)\left(1-{4999\over99997}\right)\left(1-{4999\over99996}\right)$$ $$p_2=\left(1-{5000\over100000}\right)\left({5000\over99999}\right)\left(1-{4999\over99998}\right)\left(1-{4999\over99997}\right)\left(1-{4999\over99996}\right)$$ You can see that each $p_i$ is the same, so $$p = 5p_1 \approx 0.203634$$ which is very close to the binomial approximation.
The lottery is not described explicitly, so I’ll model its relevant part by assuming that each fifths of $N=100.000$ has an equal probability $p=1/{N\choose 5}$ be chosen as high prizes. Then the probability that no number of a fifth is among $n=5000$ bought numbers is ${N-n\choose 5}p\approx 0.774$, so your probability to get at least one high prize is about $0.226$.