A box contains $8$ green and $4$ blue marbles. two marbles are selected (at once with no replacement).
Question: find the expected number of green marbles among selected ones.
The answer to is question is $4/3$. However, I got the answer $3/4$ because when you make a tree diagram, you can see that $3/4$ results have green in it since it can either be $\text{GG}$, $\text{GB}$, or $\text{BG}$.
Can anyone explain to me then why it's $4/3$ or is there just a mistake?
Best Answer
You can build the law of the random variable $X = $ number of green marbles. It is $$ \begin{array}{|c|c|c|c|} \hline X&0&1&2\cr \hline p&\frac{3}{33}&\frac{16}{33}&\frac{14}{33}\cr \hline \end{array} $$ Hence the expectation $E(X) = 0 .\frac{3}{33}+ 1.\frac{16}{33}+ 2.\frac{14}{33} = \frac{4}{3}$
Note that $$ P(X=0) = \frac{\binom{4}{2}}{\binom{12}{2}}=\frac{3}{33},\quad P(X=1) = \frac{\binom{4}{1}\binom{8}{1}}{\binom{12}{2}}=\frac{16}{33},\quad P(X=2) = \frac{\binom{8}{2}}{\binom{12}{2}} = \frac{14}{33} $$