We have a complete set of events $\{A,B,C,D\}$.
$p(A)=0.2$
$p(B)=0.3$
$p(C)=0.4$
$p(D)=0.1$
Two events happen in succession, where each event cannot occur twice. (i.e. when one happens the others' probabilities re-scale to sum up to $1$).
So we have a new complete set of events $\{AB,AC,AD,BA,BC,BD,CA,CB,CD,DA,DB,DC\}$
Since they are all mutually exclusive, the probability of, say, $B$ happening is the sum of probabilities of all events that include $B$.
If I calculate correctly, that is
$p(B)=0.608$.
Also, for $C$, it's
$p(C)=0.715$.
So if I want to know the probability of either $B$ or $C$ happening, or both, I'd sum up all the probabilities of events including $B$ or $C$ or both.
And that's $p(B+C)=0.952$.
That feels correct, but there's this formula that's valid for all events, that says $p(B+C)=1-(1-p(B))(1-p(C))$, i.e. the probability of at least one of them happening is one minus the probability of neither of them happening. But that formula yields $p(B+C)=0.888\neq 0.952$.
What am I doing wrong?
And another question - is there a way I can correctly calculate $p(B+C)$ by knowing only $p(B)=0.608$ and $p(C)=0.715$, without iterating over the whole set just to calculate $p(B+C)$ (or any other such composed event, e.g. $A+D$)?