Calculating the probability of data loss of a RAID 1+0 system
A RAID 1 system with two disks fails if both disks fails. If each disk fails with probability $p$, then this happens with probability $p^2$.

The RAID 1+0 system fails if any of the two RAID 1 component systems fails. Since each of them fails with probability $p^2$, the probability that either fails is $2p^2 - p^4$ (using inclusion-exclusion).

In symbols, let $x_1,x_2,x_3,x_4$ be the failure status of the disks, that is, $x_i$ is true if the $i$th disk failed. The entire system fails if
$$ (x_1 \land x_2) \lor (x_3 \land x_4). $$
The variables $x_1,x_2,x_3,x_4$ are independent $\mathrm{Bernoulli}(p)$ random variables. A short calculation (given above) shows that
$$ \Pr[(x_1 \land x_2) \lor (x_3 \land x_4)] = 2p^2 - p^4. $$

You can solve all similar questions in the same way.