Trying to learn Bayes' theorem with a website

[pineapplewithmouse]

And I couldn't understand the solution that the website said.
I understood everything, except one thing:
The solution stated that P(B)=P(A)*P(B∣A)+P(A doesn't happen)*P(B∣A doesn't happen).
Someone can explain me this?

 

[BigBeachBanana]

And I couldn't understand the solution that the website said.
I understood everything, except one thing:
The solution stated that P(B)=P(A)*P(B∣A)+P(A doesn't happen)*P(B∣A doesn't happen).
Someone can explain me this?

Law of Total Probability:
$$\tag{1}\Pr(B) = \Pr(B \cap A) + \Pr(B \cap A')$$
By Bayes' Theorem:
$$\tag{2}\Pr(B \cap A)=\Pr(B|A)\Pr(A)$$
Similarly,

$$\tag{3}\Pr(B \cap A')=\Pr(B|A')\Pr(A')$$
Substitute (2) and (3) into (1):
$$\Pr(B) =\Pr(B|A)\Pr(A) + \Pr(B|A')\Pr(A')$$