Intuition behind the probability of an event and another event's complement

Question

I don't get the intuition when my introductory statistics textbooks mentions this, can someone please explain it to me?:

P(D and B`) = P(D) - P(D and B)

Answer 1

$D$ occurs iff $D$ and $B$ both occur or $D$ and $B'$ both occur. This gives $P(D)=P(D\cap B) +P(D\cap B')$ and subtracting $P(D\cap B)$ form both sides gives the desired result.

Answer 2

It may be better intuitively to give an example. Suppose that D is a head and B is a tail when tossing two coins with the head a probability of $0.3$ and the tail a probability of $0.4$.

Intuitively $P(\text {D and B'}) = 0.3\cdot 0.6 = 0.18$

Using the formula in your question $P(\text{D and B'}) = P(D) - P(\text{D and B})$

$P(\text{D and B'}) = 0.3 - 0.3\cdot 0.4 = 0.3 - 0.12 = 0.18$