text: Suppose that A and B are independent events such that the probability that neither occurs is a and the probability of B is b.
  
  Show that:
  
  $\boldsymbol{ P(A) \ = \ \frac{ 1 \ – \ b \ – \ a }{ 1 \ – \ b } }$
  
  The aim of this question is to develop understanding of some of the basic probability and set theory properties for deriving some complex mathematical equations.
  
  Step 1: Given that:
  
  $P(B) \ = \ b$
  
  And:
  
  $P( \ \overline{A} \ \cap \ \overline{B} \ ) \ = \ a$
  
  Step 2: Since $A$ and $B$ are independent:
  
  $P( \ A \ \cap \ B) \ = \ P(A)P(B)$
  
  Step 3: Deriving the required expression:
  
  $P( \ \overline{A} \ \cap \ \overline{B} \ ) \ = \ a$
  
  Substituting the equation $\ \overline{A} \ \cap \ \overline{B} \ = \ \overline{A \ \cup \ B}$ in above expression:
  
  $P( \ \overline{A \ \cup \ B} \ ) \ = \ a \$
  
  Substituting the equation $\ \overline{A \ \cup \ B} \ = \ 1\ \ – \ P( \ A \ \cup \ B \ )$ in above expression:
  
  $1 \ – \ P( \ A \ \cup \ B \ ) \ = \ a$
  
  Substituting the equation $\ P( \ A \ \cup \ B \ )\ =\ P(A) \ + \ P(B) \ – \ P(A \cap B)$ in above expression:
  
  $1 \ – \ \{ \ P(A) \ + \ P(B) \ – \ P(A \cap B) \ \} \ = \ a$
  
  $1 \ – \ \ P(A) \ – \ P(B) \ + \ P(A \cap B) \ = \ a$
  
  Substituting the equation $P( \ A \ \cap \ B) \ = \ P(A) \cdot P(B)$ in above expression:
  
  $1 \ – \ \ P(A) \ – \ P(B) \ + \ P(A) \cdot P(B) \ = \ a$
  
  Substituting the equation $P(B) \ = \ b$ in above expression:
  
  $1 \ – \ \ P(A) \ – \ b \ + \ P(A) \cdot b \ = \ a$
  
  Rearranging:
  
  $1 \ – \ a \ – \ b \ = \ P(A) \ – \ P(A) \cdot b$
  
  $1 \ – \ a \ – \ b \ = \ P(A) \ ( \ 1 \ – \ b \ )$
  
  Rearranging:
  
  $P(A) \ = \ \dfrac{ 1 \ – \ a \ – \ b }{ 1 \ – \ b }$
  
  Numerical Result
  
  If $a$ is the joint probability of $A$ and $B$ not happening simultaneously and $b$ is the probability of $B$, then:
  
  $P(A) \ = \ \dfrac{ 1 \ – \ a \ – \ b }{ 1 \ – \ b }$
  
  Example
  
  If the joint probability of $A$ and $B$ not happening simultaneously is $0.2$ and the probability of $B$ is $0.1$, then find the probability of $A$.
  
  From the above derivation:
  
  $P(A) \ = \ \dfrac{ 1 \ – \ a \ – \ b }{ 1 \ – \ b }$
  
  $P(A) \ = \ \dfrac{ 1 \ – \ 0.2 \ – \ 0.1 }{ 1 \ – \ 0.1 }$
  
  $P(A) \ = \ \dfrac{ 0.7 }{ 0.9 }$
  
  $P(A) \ = \ 0.778$