I have a probability problem that I don't know if I solved it correctly. It says:
Two balls are drawn simultaneously from a box with 5 blue and 5 red balls. If they are of the same color you win 1,10 euro. Otherwise you lose 1 euro. Calculate expected value and variance.
I first build the sample space $S=\{(w_{1},w_{2})|\forall w_{1},w_{2}\in(Blue=0,Red=1)\}$
Than we are interested in the two events A={I Win$=(0,0)\cup(1,1)$} and B={I lose$={(w_{1},w_{2})|\forall w_{1},w_{2}\in(0,1)},w_{1}\neq w_{2}$}
The probability $P(A)$ is equal to: $P(A)=\frac{\binom{5}{2}*\binom{5}{0}}{\binom{10}{2}}*2=\frac{4}{9}$ and therefore we have P(B) that is $P(A)=1-P(B)=\frac{5}{9}$
The expected value is $E(X)=1,10*\frac{4}{9}-1*\frac{5}{9}=\frac{-11}{5}$ and the variance is $V(X)=\sum_{1}^{2}(x_{i}-E(x))^2*P(X=x_{i})=(1,10+\frac{11}{5})^2*\frac{4}{9}+(-1+\frac{11}{5})^2*\frac{5}{9}=5,64$ I am right or have I done some mistakes?
Your arithmetic for the expected value seems incorrect.
$E(x)=(110*4/9)+(-1*5/9) = 145/3$
This will in turn throw off your variance calculation.