Describe $\sigma$-algebra and the measure

Question

I can solve the following problem in a combinatorial way:

Suppose we have a box containing 3 white and 2 black balls. We randomly choose two balls and put them into the second box which contains 4 white and 4 black balls. Next we randomly take a ball from the second box. What is a probability that the ball is white?

But how strictly describe the probability space, the $\sigma$-algebra and the measure?

Thanks in advance.

Answer 1

There are $\binom{5}{2}$ ways two pick balls out of the first box, then $10$ ways to choose a ball from the second box. Therefore, a good choice for the probability space is any set with $\binom{5}{2}\cdot 10=100$ elements.

The $\sigma$-algebra is all subsets; as a rule of thumb, you only have to worry about the $\sigma$-algebra when the probability space is uncountable.

All events in this space are equally likely, so the probability of an event is $E$ is $|E|/|\Omega|=|E|/100$.