Probability of expected value of independent events 
Four independent astronomers are being sent into space with the
  objective of finding planet Ceres. The first astronomer has 40% chance
  of finding the planet, the second has 50% chance, the third 60% chance
  and the fourth has 80% chance. Calculated the expected value of a random variable $Z$ that is equal to the number of astronauts that found the planet.

I've tried calculating the probability of each value of the random variable by counting the possible outcomes that give such value and if more than one add them.
I ended up with something like $P(Z=1)=\frac{41}{125}$, $P(Z=2)=\frac{19}{50}$, $P(Z=3)=\frac{41}{125}$, $P(Z=4)=\frac{12}{125}$,$P(Z=0)=\frac{3}{125}$
And I proceeded on calculating the probability from there onwards. However, I tried to add the probabilities up just to confirm I'm not wrong and they don't add up to $1$. Does that mean something is wrong?

We have $Z=X_1+X_2+X_3+X_4$ where $X_1\sim\mathrm{Ber}(2/5)$, $X_2\sim\mathrm{Ber}(1/2)$, $X_3\sim\mathrm{Ber}(3/5)$, and $X_4\sim\mathrm{Ber}(4/5)$. Hence
$\begin{align*}
\mathbb E[Z] &= \mathbb E[X_1+X_2+X_3+X_4]\\ 
&= \mathbb E[X_1] + \mathbb E[X_2] + \mathbb E[X_3] + \mathbb E[X_4]\\
&= 2/5 + 1/2 + 3/5 + 4/5\\
&= 23/10. \end{align*}$
