An urn contains $4$ red balls and $6$ black balls. You draw with replacement $2$ balls.
a) Write down the sample space of this experiment and list $3$ events $A$, $B$ and $C$, whose intersection is not empty (i.e. $A ∩ B ∩ C= ∅$).
b) Let $X$ be the random variable that counts the number of red balls among those extracted. What is its distribution?
For a) The sample space is $\Omega=\{W,R\}$ , so white and red (the set whose elements describe the outcomes of the experiment), correct?
For the events in order to have intersection what should I consider? If I choose A={W,W}, B={R,R}, C={R,W} I think that should be an option, what do you think?
For b) I think that the distribution is a Binomial Distribution, it's correct?
For (a), $\Omega=\{B,R\}^2=\{BB,BR,RB,RR\}$. An example of 3 events whose intersection is not empty would be $$ A=\{BB\}, B=\{BB, BR, RB\}, C=\{BB,RR\}. $$ $A=$ "both balls are black", $B=$ "at least one ball is black", and $C=$ "both balls are of the same color".
For (b), $\mathsf{P}(X=0)=\mathsf{P}(\{BB\})=(6/10)^2$, $\mathsf{P}(X=1)=\mathsf{P}(\{BR,RB\})=2\times 24/100$, and so on. In general, $$ \mathsf{P}(X=k)=\binom{2}{k}\left(\frac{4}{10}\right)^k\left(\frac{6}{10}\right)^{2-k}. $$