Let's say a machine has $n$ balls, each with a distinct number from $1$ to $n$. The ball is chosen at random. Let's say that you come across this machine. It contains either $100$ balls or $100000$ balls. Those are the only two possibilities and they also have equal probability. Let's say you run the machine and the ball that comes out has a number less than or equal to $100$.
What is the probability that the machine contains $100$ balls?
What is the probability that the machine contains $100,000$ balls?
I made a tree diagram showing the first $2$ branches spanning to two nodes (one showing $100$ balls and another showing $100000$ balls) and each node spans into two other nodes. The $100000$ node spans out to $\leq 100$ balls (for balls with numbers marked with $\leq 100$) and $>100$ balls. The $100$ node is what I'm confused on.
The probability that the choice is $≤100$ is $1$ if you got the small one and $\frac {100}{100,000}=\frac 1{1000}$ if you got the big one. Thus the total probability of seeing a number $≤100$ is the sum $$\frac 12\times 1 +\frac 12\times \frac 1{1000}=0.5005$$ of that, $.5$ is explained by your having got the small one, so the desired probability is $$\frac {.5}{0.5005}=0.999000999$$ Unsurprisingly, it is virtually certain that you got the small one.